Post on 15-Mar-2020
HOMEWORK ASSIGNMENT 9
ACCELERATED PROOFS AND PROBLEM SOLVING [MATH08071]
Each problem will be marked out of 4 points.
Exercise 1 ([1, Exercise 20.2]). Let f and g be the following permutations in S7:
f =
(1 2 3 4 5 6 73 1 5 7 2 6 4
), g =
(1 2 3 4 5 6 73 1 7 6 4 5 2
).
Write down in cycle notation the permutations f , g, g2, g3, f ◦ g, (f ◦ g)−1, g−1 ◦ f−1.What is the order of f? What is the order of f ◦ g?
Solution. We have f = (1 3 5 2)(4 7), g = (1 3 7 2)(4 6 5), g2 = (1 7)(3 2)(4 5 6),g3 = (1 2 7 3), f ◦ g = (1 5 7)(2 3 4 6), (f ◦ g)−1 = (1 7 5)(2 6 4 3), and
g−1 ◦ f−1 = (f ◦ g)−1 = (1 7 5)(2 6 4 3).
The order of f is 4 by [1, Proposition 20.4]. Similarly, the order of f ◦ g is 12.
Exercise 2 ([1, Exercise 20.4]). A pack of 2n cards is shuffled by the “interlacing” methoddescribed in [1, Example 20.7] — in other words, if the original order is 1, 2, 3, 4, . . . , 2n,the new order after the shuffle is 1, n+1, 2, n+2, . . . , n, 2n. Work out how many times thisshuffle must be repeated before the cards are again in the original order in the followingcases:
(a) n = 10(b) n = 12(c) n = 14(d) n = 16(e) n = 24(f) n = 26 (i.e., a real pack of cards).
Investigate this question as far as you can for a general n — it is quite fascinating.
Solution. In other words, we must find the order of the permutation(1 2 3 4 5 6 · · · 2n− 1 2n1 n+ 1 2 n+ 2 3 n+ 3 · · · n 2n
)∈ S2n
for n = 10, 12, 14, 16, 24, 26. Let us find the order of the above permutation in the casesn = 10, 12, 14, 16, 24, 26 step by step. See Appendix for further investigation (for all nfrom 4 till 1000).
(10) The case n = 10. The first cycle is a cycle of length 18. This cycle looks like this:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 18, and 1. Namely, wehave the following composition of cycles:
(1)(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3)(20).
Omitting the cycles of length one, we have the following composition of cycles:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
This assignment is due on Thursday 26th November 2015.
1
So, the order of the permutation is 18 by [1, Proposition 20.4].(12) The case n = 12. The first cycle is a cycle of length 11. This cycle looks like this:
(2 13 7 4 14 19 10 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 11, 11, and 1. Namely,we have the following composition of cycles:
(1)(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11)(24).
Omitting the cycles of length one, we have the following composition of cycles:
(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11).
So, the order of the permutation is 11 by [1, Proposition 20.4].(14) The case n = 14. The first cycle is a cycle of length 18. This cycle looks like this:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3).
The permutation is a composition of 5 cycles of lengths 1, 18, 6, 2, and 1. Namely,we have the following composition of cycles:
(1)(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19)(28).
Omitting the cycles of length one, we have the following composition of cycles:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19).
So, the order of the permutation is 18 by [1, Proposition 20.4].(16) The case n = 16. The first cycle is a cycle of length 5. This cycle looks like this:
(2 17 9 5 3).
The permutation is a composition of 8 cycles of lengths 1, 5, 5, 5, 5, 5, 5, and 1.Namely, we have the following composition of cycles:
(1)(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(32).
Omitting the cycles of length one, we have the following composition of cycles:
(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31).
So, the order of the permutation is 5 by [1, Proposition 20.4].(24) The case n = 24. The first cycle is a cycle of length 23. This cycle looks like this:
(2 25 13 7 4 26 37 19 10 29 15 8 28 38 43 22 35 18 33 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 23, 23, and 1. So, theorder of the permutation is 23 by [1, Proposition 20.4].
(26) The case n = 26. The first cycle is a cycle of length 8. This cycle looks like this:
(2 27 14 33 17 9 5 3).
The permutation is a composition of 9 cycles of lengths 1, 8, 8, 8, 8, 8, 2, 8, and1. Omitting the cycles of length one, we have the following composition of cycles:
(2 27 14 33 17 9 5 3)(4 28 40 46 49 25 13 7)(6 29 15 8 30 41 21 11)
(10 31 16 34 43 22 37 19)(12 32 42 47 24 38 45 23)(18 35)(20 36 44 48 50 51 26 39).
So, the order of the permutation is 8 by [1, Proposition 20.4]. Note that 8 isrelatively small comparing to the pack size 52. I guess this is the reason why thisinterlacing trick is so popular among gamblers.
2
Exercise 3 ([1, Exercise 20.6]). This question is about the 3 × 3 version of the FifteenPizzle of Example 20.10. Starting with the configuration
1 2 34 5 67 8 �
which of the following configurations can be reached by a sequence of moves?
3 2 14 5 67 8 �
,1 � 23 4 56 7 8
,1 7 26 4 53 8 �
Solution. Arguing as in [1, Example 20.10], we see that we never can reach the configu-ration
3 2 14 5 67 8 �
,
because the corresponding permutation(1 2 3 4 5 6 7 83 2 1 4 5 6 4 8
)= (1 3)
is odd. On the other hand, we can reach the remaining two configurations
1 � 23 4 56 7 8
and1 7 26 4 53 8 �
.
Let us show this other way around. Namely, let us show how to start with any of thistwo configuration and get the initial configuration
1 2 34 5 67 8 �
.
For the first case we have
1 � 23 4 56 7 8
−→1 4 23 � 56 7 8
−→1 4 2� 3 56 7 8
−→1 4 26 3 5� 7 8
−→1 4 26 3 57 � 8
−→1 4 26 3 57 8 �
−→
−→1 4 26 3 �7 8 5
−→1 4 26 � 37 8 5
−→1 4 2� 6 37 8 5
−→1 4 27 6 3� 8 5
−→1 4 27 6 38 � 5
−→1 4 27 � 38 6 5
−→
−→1 � 27 4 38 6 5
−→1 2 �7 4 38 6 5
−→1 2 37 4 �8 6 5
−→1 2 37 4 58 6 �
−→1 2 37 4 58 � 6
−→1 2 37 4 5� 8 6
−→
−→1 2 3� 4 57 8 6
−→1 2 34 � 57 8 6
−→1 2 34 5 �7 8 6
−→1 2 34 5 67 8 �
.
For the second case we have
1 7 26 4 53 8 �
−→1 7 26 4 53 � 8
−→1 7 26 4 5� 3 8
−→1 7 2� 4 56 3 8
−→1 7 24 � 56 3 8
−→
3
−→1 7 24 3 56 � 8
−→1 7 24 3 56 8 �
−→1 7 24 3 �6 8 5
−→1 7 24 � 36 8 5
−→1 7 24 8 36 � 5
−→
−→1 7 24 8 3� 6 5
−→1 7 2� 8 34 6 5
−→1 7 28 � 34 6 5
−→1 � 28 7 34 6 5
−→1 2 �8 7 34 6 5
−→
−→1 2 38 7 �4 6 5
−→1 2 38 7 54 6 �
−→1 2 38 7 54 � 6
−→1 2 38 � 54 7 6
−→1 2 3� 8 54 7 6
−→
−→1 2 34 8 5� 7 6
−→1 2 34 8 57 � 6
−→1 2 34 � 57 8 6
−→1 2 34 5 �7 8 6
−→1 2 34 5 67 8 �
.
Now reversing the arrows, we start with the configuration
1 2 34 5 67 8 �
and get the following configurations
1 � 23 4 56 7 8
and1 7 26 4 53 8 �
.
Exercise 4 ([3, Problem 5]). Denote by α the permutation(1 2 3 4 5 6 73 7 4 1 6 5 2
)∈ S7.
(a) Express α as a product of disjoint cycles.(b) Find the order of the permutation α and find
α ◦ α ◦ α ◦ · · · ◦ α ◦ α︸ ︷︷ ︸2011 times
.
Solution. α = (1 3 4)(2 7)(5 6). Then the order of α is 6 by [1, Proposition 20.4]. Then
α ◦ α ◦ α ◦ · · · ◦ α ◦ α︸ ︷︷ ︸2011 times
= α2011 = α335×6+1 =(α6)335
α = α = (1 3 4)(2 7)(5 6).
Exercise 5 ([2, Problem 5]). Let f, g be the permutation in S7.
f =
(1 2 3 4 5 6 74 5 7 3 2 6 1
), g =
(1 2 3 4 5 6 77 6 4 5 3 2 1
).
(i) Calculate f−1 and f ◦ g.(ii) Express f as a composite of disjoint cycles.(iii) Determine whether f is an even or odd permutation.(iv) Write g as a composite of simple interchanges.
Solution. We have
f−1 =
(1 2 3 4 5 6 77 5 4 1 2 6 3
)and f ◦ g =
(1 2 3 4 5 6 71 6 3 2 7 5 4
).
We have f = (1 4 3 7)(2 5). Then f is even by [1, Proposition 20.7].4
Recall that simple interchange is just another name for a cycle of length 2. Thus,arguing as in the proof of [1, Proposition 20.6], we see that
g =
(1 2 3 4 5 6 77 6 4 5 3 2 1
)= (1 7)(2 6)(3 4 5) = (1 7)(2 6)(3 5)(3 4).
Appendix A. Interlacing cards
Let us use assumptions and notations of [1, Problem 20.4]. Let us first describe cycledecomposition for the permutation in [1, Problem 20.4] for all n from 4 to 19.
(4) The case n = 4. The permutation looks like this:(1 2 3 4 5 6 7 81 5 2 6 3 7 4 8
)The first cycle is a cycle of length 3. This cycle looks like this:
(2 5 3).
The permutation is a composition of 4 cycles of lengths 1, 3, 3, and 1. Namely, wehave the following composition of cycles:
(1)(2 5 3)(4 6 7)(8).
Omitting the cycles of length one, we have the following composition of cycles:
(2 5 3)(4 6 7).
So, the order of the permutation is 3.(5) The case n = 5. The permutation looks like this:(
1 2 3 4 5 6 7 8 9 101 6 2 7 3 8 4 9 5 10
)The first cycle is a cycle of length 6. This cycle looks like this:
(2 6 8 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 6, 2, and 1. Namely, wehave the following composition of cycles:
(1)(2 6 8 9 5 3)(4 7)(10).
Omitting the cycles of length one, we have the following composition of cycles:
(2 6 8 9 5 3)(4 7).
So, the order of the permutation is 6.(6) The case n = 6. The permutation looks like this:(
1 2 3 4 5 6 7 8 9 10 11 121 7 2 8 3 9 4 10 5 11 6 12
)The first cycle is a cycle of length 10. This cycle looks like this:
(2 7 4 8 10 11 6 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 10, and 1. Namely, wehave the following composition of cycles:
(1)(2 7 4 8 10 11 6 9 5 3)(12).
Omitting the cycles of length one, we have the following composition of cycles:
(2 7 4 8 10 11 6 9 5 3).
So, the order of the permutation is 10.5
(7) The case n = 7. The permutation looks like this:(1 2 3 4 5 6 7 8 9 10 11 12 13 141 8 2 9 3 10 4 11 5 12 6 13 7 14
)The first cycle is a cycle of length 12. This cycle looks like this:
(2 8 11 6 10 12 13 7 4 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 12, and 1. Namely, wehave the following composition of cycles:
(1)(2 8 11 6 10 12 13 7 4 9 5 3)(14).
Omitting the cycles of length one, we have the following composition of cycles:
(2 8 11 6 10 12 13 7 4 9 5 3).
So, the order of the permutation is 12.(8) The case n = 8. The permutation looks like this:(
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 9 2 10 3 11 4 12 5 13 6 14 7 15 8 16
)The first cycle is a cycle of length 4. This cycle looks like this:
(2 9 5 3).
The permutation is a composition of 6 cycles of lengths 1, 4, 4, 2, 4, and 1. Namely,we have the following composition of cycles:
(1)(2 9 5 3)(4 10 13 7)(6 11)(8 12 14 15)(16).
Omitting the cycles of length one, we have the following composition of cycles:
(2 9 5 3)(4 10 13 7)(6 11)(8 12 14 15).
So, the order of the permutation is 4.(9) The case n = 9. The first cycle is a cycle of length 8. This cycle looks like this:
(2 10 14 16 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 8, 8, and 1. Namely, wehave the following composition of cycles:
(1)(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)(18).
Omitting the cycles of length one, we have the following composition of cycles:
(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7).
So, the order of the permutation is 8.(10) The case n = 10. The first cycle is a cycle of length 18. This cycle looks like this:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 18, and 1. Namely, wehave the following composition of cycles:
(1)(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3)(20).
Omitting the cycles of length one, we have the following composition of cycles:
(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).
So, the order of the permutation is 18.6
(11) The case n = 11. The first cycle is a cycle of length 6. This cycle looks like this:
(2 12 17 9 5 3).
The permutation is a composition of 7 cycles of lengths 1, 6, 3, 6, 2, 3, and 1.Namely, we have the following composition of cycles:
(1)(2 12 17 9 5 3)(4 13 7)(6 14 18 20 21 11)(8 15)(10 16 19)(22).
Omitting the cycles of length one, we have the following composition of cycles:
(2 12 17 9 5 3)(4 13 7)(6 14 18 20 21 11)(8 15)(10 16 19).
So, the order of the permutation is 6.(12) The case n = 12. The first cycle is a cycle of length 11. This cycle looks like this:
(2 13 7 4 14 19 10 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 11, 11, and 1. Namely,we have the following composition of cycles:
(1)(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11)(24).
Omitting the cycles of length one, we have the following composition of cycles:
(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11).
So, the order of the permutation is 11.(13) The case n = 13. The first cycle is a cycle of length 20. This cycle looks like this:
(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3).
The permutation is a composition of 4 cycles of lengths 1, 20, 4, and 1. Namely,we have the following composition of cycles:
(1)(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3)(6 16 21 11)(26).
Omitting the cycles of length one, we have the following composition of cycles:
(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3)(6 16 21 11).
So, the order of the permutation is 20.(14) The case n = 14. The first cycle is a cycle of length 18. This cycle looks like this:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3).
The permutation is a composition of 5 cycles of lengths 1, 18, 6, 2, and 1. Namely,we have the following composition of cycles:
(1)(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19)(28).
Omitting the cycles of length one, we have the following composition of cycles:
(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19).
So, the order of the permutation is 18.(15) The case n = 15. The first cycle is a cycle of length 28. This cycle looks like this:
(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 28, and 1. Namely, wehave the following composition of cycles:
(1)(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3)(30).
Omitting the cycles of length one, we have the following composition of cycles:
(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3).
So, the order of the permutation is 28.7
(16) The case n = 16. The first cycle is a cycle of length 5. This cycle looks like this:
(2 17 9 5 3).
The permutation is a composition of 8 cycles of lengths 1, 5, 5, 5, 5, 5, 5, and 1.Namely, we have the following composition of cycles:
(1)(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(32).
Omitting the cycles of length one, we have the following composition of cycles:
(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31).
So, the order of the permutation is 5.(17) The case n = 17. The first cycle is a cycle of length 10. This cycle looks like this:
(2 18 26 30 32 33 17 9 5 3).
The permutation is a composition of 6 cycles of lengths 1, 10, 10, 10, 2, and 1.Namely, we have the following composition of cycles:
(1)(2 18 26 30 32 33 17 9 5 3)(4 19 10 22 28 31 16 25 13 7)(6 20 27 14 24 29 15 8 21 11)(12 23)(34).
Omitting the cycles of length one, we have the following composition of cycles:
(2 18 26 30 32 33 17 9 5 3)(4 19 10 22 28 31 16 25 13 7)(6 20 27 14 24 29 15 8 21 11)(12 23).
So, the order of the permutation is 10.(18) The case n = 18. The first cycle is a cycle of length 12. This cycle looks like this:
(2 19 10 23 12 24 30 33 17 9 5 3).
The permutation is a composition of 7 cycles of lengths 1, 12, 12, 3, 4, 3, and 1.Namely, we have the following composition of cycles:
(1)(2 19 10 23 12 24 30 33 17 9 5 3)(4 20 28 32 34 35 18 27 14 25 13 7)(6 21 11)(8 22 29 15)(16 26 31)(36).
Omitting the cycles of length one, we have the following composition of cycles:
(2 19 10 23 12 24 30 33 17 9 5 3)(4 20 28 32 34 35 18 27 14 25 13 7)(6 21 11)(8 22 29 15)(16 26 31).
So, the order of the permutation is 12.(19) The case n = 19. The first cycle is a cycle of length 36. This cycle looks like this:
(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3).
The permutation is a composition of 3 cycles of lengths 1, 36, and 1. Namely, wehave the following composition of cycles:
(1)(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3)(38).
Omitting the cycles of length one, we have the following composition of cycles:
(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3).
So, the order of the permutation is 36.
We can continue in the same fashion for a very long time (keeping in mind that most ofthis text is written by a computer program). But it would be much better to understandthe rule that stands behind this. Unfortunately, I did not find this rule (this does notmean the rule does not exists and this does not mean that it is hard to find the rule).Nevertheless, I investigated a lot of cases (using a simple program in Java). Let mesummarize my investigations in a table whose first column contains n for all n from 4 till1000, the second column contains 2n (the number of interlacing cards), the third columncontains the cycle shape of the permutation like in [1, Example 20.4] and [1, Example 20.5](but I put the cycle shape in the ascending order contrary to the book, since it looks morenatural to me), and final forth column contains the order of our permutation (what weare looking for!).
8
n 2n cycle-shape order
4 8 (12, 32) 3
5 10 (12, 2, 6) 6
6 12 (12, 10) 10
7 14 (12, 12) 12
8 16 (12, 2, 43) 4
9 18 (12, 82) 8
10 20 (12, 18) 18
11 22 (12, 2, 32, 62) 6
12 24 (12, 112) 11
13 26 (12, 4, 20) 20
14 28 (12, 2, 6, 18) 18
15 30 (12, 28) 28
16 32 (12, 56) 5
17 34 (12, 2, 103) 10
18 36 (12, 32, 4, 122) 12
19 38 (12, 36) 36
20 40 (12, 2, 123) 12
21 42 (12, 202) 20
22 44 (12, 143) 14
23 46 (12, 2, 43, 6, 122) 12
24 48 (12, 232) 23
25 50 (12, 32, 212) 21
26 52 (12, 2, 86) 8
27 54 (12, 52) 52
28 56 (12, 4, 10, 202) 20
29 58 (12, 2, 183) 18
30 60 (12, 58) 58
31 62 (12, 60) 60
32 64 (12, 2, 32, 69) 6
33 66 (12, 4, 125) 12
34 68 (12, 66) 66
9
35 70 (12, 2, 112, 222) 22
36 72 (12, 352) 35
37 74 (12, 98) 9
38 76 (12, 2, 43, 203) 20
39 78 (12, 32, 10, 302) 30
40 80 (12, 392) 39
41 82 (12, 2, 6, 18, 54) 54
42 84 (12, 82) 82
43 86 (12, 4, 810) 8
44 88 (12, 2, 283) 28
45 90 (12, 118) 11
46 92 (12, 32, 127) 12
47 94 (12, 2, 56, 106) 10
48 96 (12, 4, 18, 362) 36
49 98 (12, 482) 48
50 100 (12, 2, 6, 103, 302) 30
51 102 (12, 100) 100
52 104 (12, 512) 51
53 106 (12, 2, 32, 43, 62, 126) 12
54 108 (12, 106) 106
55 110 (12, 363) 36
56 112 (12, 2, 363) 36
57 114 (12, 284) 28
58 116 (12, 4, 112, 442) 44
59 118 (12, 2, 6, 129) 12
60 120 (12, 32, 82, 244) 24
61 122 (12, 10, 110) 110
62 124 (12, 2, 206) 20
63 126 (12, 4, 20, 100) 100
64 128 (12, 718) 7
65 130 (12, 2, 149) 14
66 132 (12, 130) 130
67 134 (12, 32, 187) 18
10
68 136 (12, 2, 43, 6, 122, 18, 362) 36
69 138 (12, 682) 68
70 140 (12, 138) 138
71 142 (12, 2, 232, 462) 46
72 144 (12, 10, 12, 602) 60
73 146 (12, 4, 285) 28
74 148 (12, 2, 32, 62, 212, 422) 42
75 150 (12, 148) 148
76 152 (12, 1510) 15
77 154 (12, 2, 6, 86, 244) 24
78 156 (12, 4, 56, 206) 20
79 158 (12, 523) 52
80 160 (12, 2, 523) 52
81 162 (12, 32, 112, 334) 33
82 164 (12, 162) 162
83 166 (12, 2, 43, 103, 206) 20
84 168 (12, 832) 83
85 170 (12, 12, 156) 156
86 172 (12, 2, 6, 189) 18
87 174 (12, 172) 172
88 176 (12, 32, 4, 122, 20, 602) 60
89 178 (12, 2, 583) 58
90 180 (12, 178) 178
91 182 (12, 180) 180
92 184 (12, 2, 603) 60
93 186 (12, 4, 365) 36
94 188 (12, 82, 10, 404) 40
95 190 (12, 2, 32, 69, 187) 18
96 192 (12, 952) 95
97 194 (12, 962) 96
98 196 (12, 2, 43, 1215) 12
99 198 (12, 196) 196
100 200 (12, 992) 99
11
101 202 (12, 2, 663) 66
102 204 (12, 32, 28, 842) 84
103 206 (12, 4, 2010) 20
104 208 (12, 2, 6, 112, 222, 662) 66
105 210 (12, 10, 18, 902) 90
106 212 (12, 210) 210
107 214 (12, 2, 352, 702) 70
108 216 (12, 4, 143, 286) 28
109 218 (12, 32, 56, 1512) 15
110 220 (12, 2, 98, 188) 18
111 222 (12, 82, 12, 248) 24
112 224 (12, 376) 37
113 226 (12, 2, 43, 6, 122, 203, 602) 60
114 228 (12, 226) 226
115 230 (12, 763) 76
116 232 (12, 2, 32, 62, 103, 306) 30
117 234 (12, 298) 29
118 236 (12, 4, 232, 922) 92
119 238 (12, 2, 392, 782) 78
120 240 (12, 1192) 119
121 242 (12, 2410) 24
122 244 (12, 2, 6, 18, 54, 162) 162
123 246 (12, 32, 4, 122, 212, 842) 84
124 248 (12, 12, 18, 366) 36
125 250 (12, 2, 823) 82
126 252 (12, 505) 50
127 254 (12, 10, 112, 1102) 110
128 256 (12, 2, 43, 830) 8
129 258 (12, 1616) 16
130 260 (12, 32, 367) 36
131 262 (12, 2, 6, 283, 842) 84
132 264 (12, 1312) 131
133 266 (12, 4, 525) 52
12
134 268 (12, 2, 118, 228) 22
135 270 (12, 268) 268
136 272 (12, 1352) 135
137 274 (12, 2, 32, 62, 1221) 12
138 276 (12, 4, 10, 2013) 20
139 278 (12, 923) 92
140 280 (12, 2, 56, 6, 106, 306) 30
141 282 (12, 704) 70
142 284 (12, 943) 94
143 286 (12, 2, 43, 183, 366) 36
144 288 (12, 32, 202, 604) 60
145 290 (12, 82, 1362) 136
146 292 (12, 2, 486) 48
147 294 (12, 292) 292
148 296 (12, 4, 58, 1162) 116
149 298 (12, 2, 6, 103, 18, 302, 902) 90
150 300 (12, 112, 12, 1322) 132
151 302 (12, 32, 143, 426) 42
152 304 (12, 2, 1003) 100
153 306 (12, 4, 605) 60
154 308 (12, 1023) 102
155 310 (12, 2, 512, 1022) 102
156 312 (12, 1552) 155
157 314 (12, 1562) 156
158 316 (12, 2, 32, 43, 69, 1220) 12
159 318 (12, 316) 316
160 320 (12, 10, 28, 1402) 140
161 322 (12, 2, 1063) 106
162 324 (12, 82, 18, 724) 72
163 326 (12, 4, 125, 20, 604) 60
164 328 (12, 2, 369) 36
165 330 (12, 32, 232, 694) 69
166 332 (12, 3011) 30
13
167 334 (12, 2, 6, 369) 36
168 336 (12, 4, 66, 1322) 132
169 338 (12, 2116) 21
170 340 (12, 2, 2812) 28
171 342 (12, 56, 1031) 10
172 344 (12, 32, 212, 1472) 147
173 346 (12, 2, 43, 112, 222, 446) 44
174 348 (12, 346) 346
175 350 (12, 348) 348
176 352 (12, 2, 6, 129, 18, 366) 36
177 354 (12, 884) 88
178 356 (12, 4, 352, 1402) 140
179 358 (12, 2, 32, 62, 86, 2412) 24
180 360 (12, 1792) 179
181 362 (12, 18, 342) 342
182 364 (12, 2, 103, 1103) 110
183 366 (12, 4, 98, 368) 36
184 368 (12, 1832) 183
185 370 (12, 2, 6, 206, 604) 60
186 372 (12, 32, 52, 1562) 156
187 374 (12, 372) 372
188 376 (12, 2, 43, 203, 1003) 100
189 378 (12, 12, 28, 844) 84
190 380 (12, 378) 378
191 382 (12, 2, 718, 1418) 14
192 384 (12, 1912) 191
193 386 (12, 32, 4, 10, 122, 202, 302, 604) 60
194 388 (12, 2, 6, 149, 426) 42
195 390 (12, 388) 388
196 392 (12, 82, 112, 884) 88
197 394 (12, 2, 1303) 130
198 396 (12, 4, 392, 1562) 156
199 398 (12, 449) 44
14
200 400 (12, 2, 32, 62, 1821) 18
201 402 (12, 2002) 200
202 404 (12, 56, 12, 606) 60
203 406 (12, 2, 43, 6, 122, 18, 362, 54, 1082) 108
204 408 (12, 10, 36, 1802) 180
205 410 (12, 2042) 204
206 412 (12, 2, 686) 68
207 414 (12, 32, 58, 1742) 174
208 416 (12, 4, 82, 1642) 164
209 418 (12, 2, 1383) 138
210 420 (12, 418) 418
211 422 (12, 420) 420
212 424 (12, 2, 6, 232, 462, 1382) 138
213 426 (12, 4, 810, 20, 408) 40
214 428 (12, 32, 607) 60
215 430 (12, 2, 103, 123, 606) 60
216 432 (12, 4310) 43
217 434 (12, 726) 72
218 436 (12, 2, 43, 2815) 28
219 438 (12, 112, 18, 1982) 198
220 440 (12, 736) 73
221 442 (12, 2, 32, 69, 212, 428) 42
222 444 (12, 442) 442
223 446 (12, 4, 118, 448) 44
224 448 (12, 2, 1483) 148
225 450 (12, 2242) 224
226 452 (12, 10, 2022) 20
227 454 (12, 2, 1510, 3010) 30
228 456 (12, 32, 4, 1237) 12
229 458 (12, 766) 76
230 460 (12, 2, 6, 86, 18, 244, 724) 72
231 462 (12, 460) 460
232 464 (12, 2312) 231
15
233 466 (12, 2, 43, 56, 106, 2018) 20
234 468 (12, 466) 466
235 470 (12, 32, 667) 66
236 472 (12, 2, 529) 52
237 474 (12, 10, 143, 706) 70
238 476 (12, 4, 18, 20, 362, 1802) 180
239 478 (12, 2, 6, 523, 1562) 156
240 480 (12, 2392) 239
241 482 (12, 12, 3613) 36
242 484 (12, 2, 32, 62, 112, 222, 334, 664) 66
243 486 (12, 4, 4810) 48
244 488 (12, 2432) 243
245 490 (12, 2, 1623) 162
246 492 (12, 490) 490
247 494 (12, 82, 28, 568) 56
248 496 (12, 2, 43, 6, 103, 122, 206, 302, 604) 60
249 498 (12, 32, 352, 1054) 105
250 500 (12, 1663) 166
251 502 (12, 2, 832, 1662) 166
252 504 (12, 2512) 251
253 506 (12, 4, 1005) 100
254 508 (12, 2, 123, 1563) 156
255 510 (12, 508) 508
256 512 (12, 32, 956) 9
257 514 (12, 2, 6, 1828) 18
258 516 (12, 4, 512, 2042) 204
259 518 (12, 10, 232, 2302) 230
260 520 (12, 2, 1723) 172
261 522 (12, 2602) 260
262 524 (12, 522) 522
263 526 (12, 2, 32, 43, 62, 126, 203, 606) 60
264 528 (12, 56, 82, 4012) 40
265 530 (12, 112, 2532) 253
16
266 532 (12, 2, 6, 583, 1742) 174
267 534 (12, 12, 202, 608) 60
268 536 (12, 4, 106, 2122) 212
269 538 (12, 2, 1783) 178
270 540 (12, 32, 10, 212, 302, 2102) 210
271 542 (12, 540) 540
272 544 (12, 2, 1803) 180
273 546 (12, 4, 3615) 36
274 548 (12, 546) 546
275 550 (12, 2, 6, 609) 60
276 552 (12, 18, 28, 2522) 252
277 554 (12, 32, 3914) 39
278 556 (12, 2, 43, 3615) 36
279 558 (12, 556) 556
280 560 (12, 12, 143, 846) 84
281 562 (12, 2, 86, 103, 4012) 40
282 564 (12, 562) 562
283 566 (12, 4, 2820) 28
284 568 (12, 2, 32, 69, 187, 547) 54
285 570 (12, 2842) 284
286 572 (12, 1145) 114
287 574 (12, 2, 952, 1902) 190
288 576 (12, 4, 112, 20, 442, 2202) 220
289 578 (12, 1444) 144
290 580 (12, 2, 966) 96
291 582 (12, 32, 82, 2462) 246
292 584 (12, 10, 52, 2602) 260
293 586 (12, 2, 43, 6, 1247) 12
294 588 (12, 586) 586
295 590 (12, 56, 18, 906) 90
296 592 (12, 2, 1963) 196
297 594 (12, 1484) 148
298 596 (12, 32, 4, 810, 122, 2420) 24
17
299 598 (12, 2, 992, 1982) 198
300 600 (12, 2992) 299
301 602 (12, 2524) 25
302 604 (12, 2, 6, 669) 66
303 606 (12, 4, 10, 202, 110, 2202) 220
304 608 (12, 3032) 303
305 610 (12, 2, 32, 62, 283, 846) 84
306 612 (12, 12, 232, 2762) 276
307 614 (12, 612) 612
308 616 (12, 2, 43, 2030) 20
309 618 (12, 1544) 154
310 620 (12, 618) 618
311 622 (12, 2, 6, 112, 18, 222, 662, 1982) 198
312 624 (12, 32, 118, 3316) 33
313 626 (12, 4, 20, 100, 500) 500
314 628 (12, 2, 103, 183, 906) 90
315 630 (12, 82, 36, 728) 72
316 632 (12, 4514) 45
317 634 (12, 2, 2103) 210
318 636 (12, 4, 718, 2818) 28
319 638 (12, 32, 127, 212, 846) 84
320 640 (12, 2, 6, 352, 702, 2102) 210
321 642 (12, 6410) 64
322 644 (12, 2143) 214
323 646 (12, 2, 43, 149, 2818) 28
324 648 (12, 3232) 323
325 650 (12, 10, 58, 2902) 290
326 652 (12, 2, 32, 56, 62, 106, 1512, 3012) 30
327 654 (12, 652) 652
328 656 (12, 4, 130, 2602) 260
329 658 (12, 2, 6, 98, 1832) 18
330 660 (12, 658) 658
331 662 (12, 660) 660
18
332 664 (12, 2, 86, 123, 2424) 24
333 666 (12, 32, 4, 122, 187, 3614) 36
334 668 (12, 112, 28, 3082) 308
335 670 (12, 2, 376, 746) 74
336 672 (12, 10, 6011) 60
337 674 (12, 4814) 48
338 676 (12, 2, 43, 6, 122, 18, 203, 362, 602, 1802) 180
339 678 (12, 676) 676
340 680 (12, 32, 4814) 48
341 682 (12, 2, 2263) 226
342 684 (12, 2231) 22
343 686 (12, 4, 6810) 68
344 688 (12, 2, 769) 76
345 690 (12, 12, 52, 1564) 156
346 692 (12, 2303) 230
347 694 (12, 2, 32, 69, 103, 3020) 30
348 696 (12, 4, 138, 2762) 276
349 698 (12, 82, 202, 4016) 40
350 700 (12, 2, 298, 588) 58
351 702 (12, 700) 700
352 704 (12, 18, 3619) 36
353 706 (12, 2, 43, 232, 462, 926) 92
354 708 (12, 32, 100, 3002) 300
355 710 (12, 708) 708
356 712 (12, 2, 6, 392, 788) 78
357 714 (12, 56, 112, 5512) 55
358 716 (12, 4, 10, 125, 202, 6010) 60
359 718 (12, 2, 1192, 2382) 238
360 720 (12, 3592) 359
361 722 (12, 32, 5114) 51
362 724 (12, 2, 2430) 24
363 726 (12, 4, 20, 285, 1404) 140
364 728 (12, 1216) 121
19
365 730 (12, 2, 6, 18, 54, 162, 486) 486
366 732 (12, 82, 143, 5612) 56
367 734 (12, 2443) 244
368 736 (12, 2, 32, 43, 62, 126, 212, 422, 846) 84
369 738 (12, 10, 66, 3302) 330
370 740 (12, 2463) 246
371 742 (12, 2, 123, 183, 3618) 36
372 744 (12, 3712) 371
373 746 (12, 4, 1485) 148
374 748 (12, 2, 6, 823, 2462) 246
375 750 (12, 32, 106, 3182) 318
376 752 (12, 3752) 375
377 754 (12, 2, 5015) 50
378 756 (12, 4, 1510, 6010) 60
379 758 (12, 756) 756
380 760 (12, 2, 103, 112, 222, 1106) 110
381 762 (12, 3802) 380
382 764 (12, 32, 3621) 36
383 766 (12, 2, 43, 6, 830, 122, 2420) 24
384 768 (12, 12, 58, 3482) 348
385 770 (12, 3842) 384
386 772 (12, 2, 1648) 16
387 774 (12, 772) 772
388 776 (12, 4, 56, 2037) 20
389 778 (12, 2, 32, 62, 3621) 36
390 780 (12, 18, 202, 1804) 180
391 782 (12, 10, 352, 7010) 70
392 784 (12, 2, 6, 18, 283, 842, 2522) 252
393 786 (12, 4, 5215) 52
394 788 (12, 786) 786
395 790 (12, 2, 1312, 2622) 262
396 792 (12, 32, 284, 848) 84
397 794 (12, 12, 6013) 60
20
398 796 (12, 2, 43, 5215) 52
399 798 (12, 796) 796
400 800 (12, 82, 232, 1844) 184
401 802 (12, 2, 6, 118, 228, 668) 66
402 804 (12, 98, 10, 908) 90
403 806 (12, 32, 4, 112, 122, 334, 442, 1324) 132
404 808 (12, 2, 2683) 268
405 810 (12, 4042) 404
406 812 (12, 2703) 270
407 814 (12, 2, 1352, 2702) 270
408 816 (12, 4, 162, 3242) 324
409 818 (12, 143, 18, 1266) 126
410 820 (12, 2, 32, 69, 1263) 12
411 822 (12, 820) 820
412 824 (12, 4112) 411
413 826 (12, 2, 43, 103, 2039) 20
414 828 (12, 826) 826
415 830 (12, 828) 828
416 832 (12, 2, 929) 92
417 834 (12, 32, 82, 212, 244, 1684) 168
418 836 (12, 4, 832, 3322) 332
419 838 (12, 2, 56, 6, 106, 18, 306, 906) 90
420 840 (12, 4192) 419
421 842 (12, 28, 812) 812
422 844 (12, 2, 7012) 70
423 846 (12, 4, 125, 1565) 156
424 848 (12, 32, 10, 302, 110, 3302) 330
425 850 (12, 2, 949) 94
426 852 (12, 112, 36, 3962) 396
427 854 (12, 852) 852
428 856 (12, 2, 43, 6, 122, 189, 3618) 36
429 858 (12, 4282) 428
430 860 (12, 858) 858
21
431 862 (12, 2, 32, 62, 206, 6012) 60
432 864 (12, 4312) 431
433 866 (12, 4, 1725) 172
434 868 (12, 2, 86, 1366) 136
435 870 (12, 10, 392, 3902) 390
436 872 (12, 12, 66, 1326) 132
437 874 (12, 2, 6, 4818) 48
438 876 (12, 32, 4, 122, 20, 602, 100, 3002) 300
439 878 (12, 876) 876
440 880 (12, 2, 2923) 292
441 882 (12, 5516) 55
442 884 (12, 882) 882
443 886 (12, 2, 43, 583, 1166) 116
444 888 (12, 4432) 443
445 890 (12, 32, 718, 2136) 21
446 892 (12, 2, 6, 103, 18, 302, 54, 902, 2702) 270
447 894 (12, 18, 232, 4142) 414
448 896 (12, 4, 178, 3562) 356
449 898 (12, 2, 112, 123, 222, 1326) 132
450 900 (12, 56, 28, 1406) 140
451 902 (12, 82, 52, 1048) 104
452 904 (12, 2, 32, 62, 149, 4218) 42
453 906 (12, 4, 1805) 180
454 908 (12, 906) 906
455 910 (12, 2, 6, 1003, 3002) 300
456 912 (12, 9110) 91
457 914 (12, 10, 82, 4102) 410
458 916 (12, 2, 43, 6015) 60
459 918 (12, 32, 130, 3902) 390
460 920 (12, 1536) 153
461 922 (12, 2, 1029) 102
462 924 (12, 12, 352, 4202) 420
463 926 (12, 4, 20, 365, 1804) 180
22
464 928 (12, 2, 6, 512, 1028) 102
465 930 (12, 4642) 464
466 932 (12, 32, 187, 212, 1266) 126
467 934 (12, 2, 1552, 3102) 310
468 936 (12, 4, 810, 10, 202, 4020) 40
469 938 (12, 1178) 117
470 940 (12, 2, 1566) 156
471 942 (12, 940) 940
472 944 (12, 112, 202, 2204) 220
473 946 (12, 2, 32, 43, 69, 1220, 187, 3614) 36
474 948 (12, 946) 946
475 950 (12, 98, 12, 3624) 36
476 952 (12, 2, 3163) 316
477 954 (12, 6814) 68
478 956 (12, 4, 952, 3802) 380
479 958 (12, 2, 103, 283, 1406) 140
480 960 (12, 32, 682, 2044) 204
481 962 (12, 56, 1556) 155
482 964 (12, 2, 6, 1063, 3182) 318
483 966 (12, 4, 9610) 96
484 968 (12, 4832) 483
485 970 (12, 2, 86, 183, 7212) 72
486 972 (12, 1945) 194
487 974 (12, 32, 1387) 138
488 976 (12, 2, 43, 1215, 203, 6012) 60
489 978 (12, 4882) 488
490 980 (12, 10, 118, 1108) 110
491 982 (12, 2, 6, 3627) 36
492 984 (12, 4912) 491
493 986 (12, 4, 1965) 196
494 988 (12, 2, 32, 62, 232, 462, 694, 1384) 138
495 990 (12, 112, 143, 1546) 154
496 992 (12, 4952) 495
23
497 994 (12, 2, 3033) 30
498 996 (12, 4, 992, 3962) 396
499 998 (12, 3323) 332
500 1000 (12, 2, 6, 18, 3627) 36
501 1002 (12, 32, 10, 127, 302, 6014) 60
502 1004 (12, 82, 58, 2324) 232
503 1006 (12, 2, 43, 663, 1326) 132
504 1008 (12, 18, 52, 4682) 468
505 1010 (12, 5042) 504
506 1012 (12, 2, 2116, 4216) 42
507 1014 (12, 9211) 92
508 1016 (12, 32, 4, 122, 285, 8410) 84
509 1018 (12, 2, 6, 2812, 848) 84
510 1020 (12, 1018) 1018
511 1022 (12, 3403) 340
512 1024 (12, 2, 56, 1099) 10
513 1026 (12, 4, 2051) 20
514 1028 (12, 12, 392, 1566) 156
515 1030 (12, 2, 32, 62, 212, 422, 1472, 2942) 294
516 1032 (12, 5152) 515
517 1034 (12, 2584) 258
518 1036 (12, 2, 43, 6, 112, 122, 222, 446, 662, 1324) 132
519 1038 (12, 82, 60, 1208) 120
520 1040 (12, 5192) 519
521 1042 (12, 2, 3463) 346
522 1044 (12, 32, 148, 4442) 444
523 1046 (12, 4, 10, 18, 202, 362, 902, 1804) 180
524 1048 (12, 2, 3483) 348
525 1050 (12, 2624) 262
526 1052 (12, 3503) 350
527 1054 (12, 2, 6, 129, 18, 366, 54, 1086) 108
528 1056 (12, 4, 210, 4202) 420
529 1058 (12, 32, 1570) 15
24
530 1060 (12, 2, 8812) 88
531 1062 (12, 1060) 1060
532 1064 (12, 5312) 531
533 1066 (12, 2, 43, 352, 702, 1406) 140
534 1068 (12, 10, 482, 2404) 240
535 1070 (12, 3563) 356
536 1072 (12, 2, 32, 69, 86, 2440) 24
537 1074 (12, 28, 36, 2524) 252
538 1076 (12, 4, 143, 20, 286, 1406) 140
539 1078 (12, 2, 1792, 3582) 358
540 1080 (12, 12, 82, 4922) 492
541 1082 (12, 112, 232, 2534) 253
542 1084 (12, 2, 183, 3423) 342
543 1086 (12, 32, 4, 56, 122, 1512, 206, 6012) 60
544 1088 (12, 5432) 543
545 1090 (12, 2, 6, 103, 302, 1103, 3302) 330
546 1092 (12, 1090) 1090
547 1094 (12, 3643) 364
548 1096 (12, 2, 43, 98, 188, 3624) 36
549 1098 (12, 2744) 274
550 1100 (12, 32, 523, 1566) 156
551 1102 (12, 2, 1832, 3662) 366
552 1104 (12, 2938) 29
553 1106 (12, 4, 810, 125, 2440) 24
554 1108 (12, 2, 6, 18, 206, 604, 1804) 180
555 1110 (12, 1108) 1108
556 1112 (12, 10, 10011) 100
557 1114 (12, 2, 32, 62, 523, 1566) 156
558 1116 (12, 4, 376, 1486) 148
559 1118 (12, 1116) 1116
560 1120 (12, 2, 3723) 372
561 1122 (12, 18, 58, 5222) 522
562 1124 (12, 1122) 1122
25
563 1126 (12, 2, 43, 6, 122, 203, 602, 1003, 3002) 300
564 1128 (12, 32, 112, 212, 334, 2314) 231
565 1130 (12, 5642) 564
566 1132 (12, 2, 123, 283, 8412) 84
567 1134 (12, 10, 512, 5102) 510
568 1136 (12, 4, 226, 4522) 452
569 1138 (12, 2, 3783) 378
570 1140 (12, 82, 66, 2644) 264
571 1142 (12, 32, 1627) 162
572 1144 (12, 2, 6, 718, 1418, 4218) 42
573 1146 (12, 4, 7615) 76
574 1148 (12, 56, 36, 1806) 180
575 1150 (12, 2, 1912, 3822) 382
576 1152 (12, 5752) 575
577 1154 (12, 2884) 288
578 1156 (12, 2, 32, 43, 62, 103, 126, 206, 306, 6012) 60
579 1158 (12, 118, 12, 1328) 132
580 1160 (12, 18, 60, 1806) 180
581 1162 (12, 2, 6, 149, 18, 426, 1266) 126
582 1164 (12, 1667) 166
583 1166 (12, 4, 298, 1168) 116
584 1168 (12, 2, 3883) 388
585 1170 (12, 32, 832, 2494) 249
586 1172 (12, 1170) 1170
587 1174 (12, 2, 86, 112, 222, 8812) 88
588 1176 (12, 4, 20, 232, 922, 4602) 460
589 1178 (12, 10, 106, 5302) 530
590 1180 (12, 2, 6, 1303, 3902) 390
591 1182 (12, 2365) 236
592 1184 (12, 32, 127, 1567) 156
593 1186 (12, 2, 43, 392, 782, 1566) 156
594 1188 (12, 1186) 1186
595 1190 (12, 202, 28, 1408) 140
26
596 1192 (12, 2, 4427) 44
597 1194 (12, 2984) 298
598 1196 (12, 4, 1192, 4762) 476
599 1198 (12, 2, 32, 69, 1863) 18
600 1200 (12, 10, 363, 1806) 180
601 1202 (12, 3004) 300
602 1204 (12, 2, 2006) 200
603 1206 (12, 4, 2450) 24
604 1208 (12, 82, 352, 2804) 280
605 1210 (12, 2, 56, 106, 123, 6018) 60
606 1212 (12, 32, 172, 5162) 516
607 1214 (12, 1212) 1212
608 1216 (12, 2, 43, 6, 122, 18, 362, 54, 1082, 162, 3242) 324
609 1218 (12, 1528) 152
610 1220 (12, 112, 52, 5722) 572
611 1222 (12, 2, 103, 363, 1806) 180
612 1224 (12, 6112) 611
613 1226 (12, 32, 4, 122, 20, 212, 602, 842, 4202) 420
614 1228 (12, 2, 2046) 204
615 1230 (12, 1228) 1228
616 1232 (12, 6152) 615
617 1234 (12, 2, 6, 686, 2044) 204
618 1236 (12, 4, 125, 18, 3632) 36
619 1238 (12, 1236) 1236
620 1240 (12, 2, 32, 62, 583, 1746) 174
621 1242 (12, 82, 98, 7216) 72
622 1244 (12, 10, 284, 1408) 140
623 1246 (12, 2, 43, 823, 1646) 164
624 1248 (12, 143, 2843) 28
625 1250 (12, 1568) 156
626 1252 (12, 2, 6, 1389) 138
627 1254 (12, 32, 178, 5342) 534
628 1256 (12, 4, 505, 10010) 100
27
629 1258 (12, 2, 4183) 418
630 1260 (12, 1258) 1258
631 1262 (12, 12, 4826) 48
632 1264 (12, 2, 4203) 420
633 1266 (12, 4, 10, 112, 202, 442, 1102, 2204) 220
634 1268 (12, 32, 1807) 180
635 1270 (12, 2, 6, 18, 232, 462, 1382, 4142) 414
636 1272 (12, 56, 2062) 20
637 1274 (12, 18, 66, 1986) 198
638 1276 (12, 2, 43, 830, 203, 4024) 40
639 1278 (12, 1276) 1276
640 1280 (12, 6392) 639
641 1282 (12, 2, 32, 62, 6021) 60
642 1284 (12, 1282) 1282
643 1286 (12, 4, 1680) 16
644 1288 (12, 2, 6, 103, 129, 302, 6018) 60
645 1290 (12, 1618) 161
646 1292 (12, 1290) 1290
647 1294 (12, 2, 4310, 8610) 86
648 1296 (12, 32, 4, 122, 3635) 36
649 1298 (12, 6482) 648
650 1300 (12, 2, 7218) 72
651 1302 (12, 1300) 1300
652 1304 (12, 6512) 651
653 1306 (12, 2, 43, 6, 122, 2815, 8410) 84
654 1308 (12, 1306) 1306
655 1310 (12, 32, 82, 10, 244, 302, 404, 1208) 120
656 1312 (12, 2, 112, 183, 222, 1986) 198
657 1314 (12, 12, 100, 3004) 300
658 1316 (12, 4, 1312, 5242) 524
659 1318 (12, 2, 736, 1466) 146
660 1320 (12, 6592) 659
661 1322 (12, 6022) 60
28
662 1324 (12, 2, 32, 69, 187, 212, 428, 1266) 126
663 1326 (12, 4, 20, 525, 2604) 260
664 1328 (12, 2216) 221
665 1330 (12, 2, 4423) 442
666 1332 (12, 10, 110, 1210) 1210
667 1334 (12, 56, 143, 7018) 70
668 1336 (12, 2, 43, 118, 228, 4424) 44
669 1338 (12, 32, 952, 2854) 285
670 1340 (12, 12, 512, 2046) 204
671 1342 (12, 2, 6, 1483, 4442) 444
672 1344 (12, 82, 392, 3124) 312
673 1346 (12, 4, 2685) 268
674 1348 (12, 2, 2246) 224
675 1350 (12, 18, 352, 6302) 630
676 1352 (12, 32, 9614) 96
677 1354 (12, 2, 103, 2066) 20
678 1356 (12, 4, 1352, 5402) 540
679 1358 (12, 112, 58, 6382) 638
680 1360 (12, 2, 6, 1510, 3040) 30
681 1362 (12, 6802) 680
682 1364 (12, 232, 28, 6442) 644
683 1366 (12, 2, 32, 43, 62, 12111) 12
684 1368 (12, 6832) 683
685 1370 (12, 36, 1332) 1332
686 1372 (12, 2, 7618) 76
687 1374 (12, 1372) 1372
688 1376 (12, 4, 10, 2013, 10011) 100
689 1378 (12, 2, 6, 86, 18, 244, 54, 724, 2164) 216
690 1380 (12, 32, 196, 5882) 588
691 1382 (12, 1380) 1380
692 1384 (12, 2, 4603) 460
693 1386 (12, 4, 9215) 92
694 1388 (12, 98, 1873) 18
29
695 1390 (12, 2, 2312, 4622) 462
696 1392 (12, 12, 106, 6362) 636
697 1394 (12, 32, 9914) 99
698 1396 (12, 2, 43, 56, 6, 106, 122, 2018, 306, 6012) 60
699 1398 (12, 718, 10, 7018) 70
700 1400 (12, 2336) 233
701 1402 (12, 2, 4663) 466
702 1404 (12, 112, 60, 6602) 660
703 1406 (12, 4, 704, 1408) 140
704 1408 (12, 2, 32, 62, 6621) 66
705 1410 (12, 7042) 704
706 1412 (12, 82, 82, 3284) 328
707 1414 (12, 2, 6, 529, 1566) 156
708 1416 (12, 4, 943, 1886) 188
709 1418 (12, 12, 3639) 36
710 1420 (12, 2, 103, 149, 7018) 70
711 1422 (12, 32, 212, 28, 8416) 84
712 1424 (12, 2376) 237
713 1426 (12, 2, 43, 183, 203, 366, 1806) 180
714 1428 (12, 1426) 1426
715 1430 (12, 8417) 84
716 1432 (12, 2, 6, 18, 523, 1562, 4682) 468
717 1434 (12, 1798) 179
718 1436 (12, 32, 4, 122, 2010, 6020) 60
719 1438 (12, 2, 2392, 4782) 478
720 1440 (12, 7192) 719
721 1442 (12, 10, 13011) 130
722 1444 (12, 2, 123, 3639) 36
723 1446 (12, 4, 810, 13610) 136
724 1448 (12, 7232) 723
725 1450 (12, 2, 32, 69, 112, 222, 334, 6618) 66
726 1452 (12, 1450) 1450
727 1454 (12, 1452) 1452
30
728 1456 (12, 2, 43, 4830) 48
729 1458 (12, 56, 232, 11512) 115
730 1460 (12, 4863) 486
731 1462 (12, 2, 2432, 4862) 486
732 1464 (12, 32, 10, 187, 302, 9014) 90
733 1466 (12, 4, 2925) 292
734 1468 (12, 2, 6, 1629) 162
735 1470 (12, 12, 284, 8416) 84
736 1472 (12, 2456) 245
737 1474 (12, 2, 4903) 490
738 1476 (12, 4, 20, 58, 1162, 5802) 580
739 1478 (12, 32, 2107) 210
740 1480 (12, 2, 86, 283, 5624) 56
741 1482 (12, 3704) 370
742 1484 (12, 1482) 1482
743 1486 (12, 2, 43, 6, 103, 122, 18, 206, 302, 362, 604, 902, 1804) 180
744 1488 (12, 7432) 743
745 1490 (12, 7442) 744
746 1492 (12, 2, 32, 62, 352, 702, 1054, 2104) 210
747 1494 (12, 1492) 1492
748 1496 (12, 4, 112, 125, 442, 13210) 132
749 1498 (12, 2, 1669) 166
750 1500 (12, 1498) 1498
751 1502 (12, 18, 392, 2346) 234
752 1504 (12, 2, 6, 832, 1662, 4982) 498
753 1506 (12, 32, 4, 122, 143, 286, 426, 8412) 84
754 1508 (12, 10, 682, 3404) 340
755 1510 (12, 2, 2512, 5022) 502
756 1512 (12, 7552) 755
757 1514 (12, 82, 118, 8816) 88
758 1516 (12, 2, 43, 10015) 100
759 1518 (12, 202, 36, 1808) 180
760 1520 (12, 32, 56, 1512, 212, 10512) 105
31
761 1522 (12, 2, 6, 129, 1569) 156
762 1524 (12, 1522) 1522
763 1526 (12, 4, 20, 6025) 60
764 1528 (12, 2, 5083) 508
765 1530 (12, 10, 138, 6902) 690
766 1532 (12, 1530) 1530
767 1534 (12, 2, 32, 62, 956, 1856) 18
768 1536 (12, 4, 1023, 2046) 204
769 1538 (12, 28, 52, 3644) 364
770 1540 (12, 2, 6, 1828, 5419) 54
771 1542 (12, 112, 6623) 66
772 1544 (12, 7712) 771
773 1546 (12, 2, 43, 512, 1022, 2046) 204
774 1548 (12, 32, 82, 127, 2460) 24
775 1550 (12, 1548) 1548
776 1552 (12, 2, 103, 232, 462, 2306) 230
777 1554 (12, 1948) 194
778 1556 (12, 4, 1552, 6202) 620
779 1558 (12, 2, 6, 1723, 5162) 516
780 1560 (12, 7792) 779
781 1562 (12, 32, 376, 11112) 111
782 1564 (12, 2, 2606) 260
783 1566 (12, 4, 15610) 156
784 1568 (12, 7832) 783
785 1570 (12, 2, 5223) 522
786 1572 (12, 1570) 1570
787 1574 (12, 10, 12, 602, 110, 6602) 660
788 1576 (12, 2, 32, 43, 69, 1220, 203, 6020) 60
789 1578 (12, 18, 82, 7382) 738
790 1580 (12, 5263) 526
791 1582 (12, 2, 56, 86, 106, 4036) 40
792 1584 (12, 7912) 791
793 1586 (12, 4, 3165) 316
32
794 1588 (12, 2, 112, 222, 2532, 5062) 506
795 1590 (12, 32, 226, 6782) 678
796 1592 (12, 143, 36, 2526) 252
797 1594 (12, 2, 6, 18, 583, 1742, 5222) 522
798 1596 (12, 4, 10, 202, 285, 14010) 140
799 1598 (12, 5323) 532
800 1600 (12, 2, 123, 206, 6024) 60
801 1602 (12, 4004) 400
802 1604 (12, 32, 763, 2286) 228
803 1606 (12, 2, 43, 1063, 2126) 212
804 1608 (12, 8032) 803
805 1610 (12, 2018) 201
806 1612 (12, 2, 6, 1783, 5342) 534
807 1614 (12, 5231) 52
808 1616 (12, 4, 810, 18, 362, 7220) 72
809 1618 (12, 2, 32, 62, 103, 212, 306, 422, 2106) 210
810 1620 (12, 1618) 1618
811 1622 (12, 1620) 1620
812 1624 (12, 2, 5403) 540
813 1626 (12, 4, 125, 20, 604, 100, 3004) 300
814 1628 (12, 5423) 542
815 1630 (12, 2, 6, 1809) 180
816 1632 (12, 32, 298, 8716) 87
817 1634 (12, 112, 352, 3854) 385
818 1636 (12, 2, 43, 3645) 36
819 1638 (12, 1636) 1636
820 1640 (12, 10, 148, 7402) 740
821 1642 (12, 2, 5463) 546
822 1644 (12, 56, 52, 2606) 260
823 1646 (12, 32, 4, 122, 232, 694, 922, 2764) 276
824 1648 (12, 2, 6, 18, 609, 1806) 180
825 1650 (12, 82, 4834) 48
826 1652 (12, 718, 12, 8418) 84
33
827 1654 (12, 2, 183, 283, 2526) 252
828 1656 (12, 4, 3011, 6022) 60
829 1658 (12, 9218) 92
830 1660 (12, 2, 32, 62, 3914, 7814) 78
831 1662 (12, 10, 1510, 3050) 30
832 1664 (12, 8312) 831
833 1666 (12, 2, 43, 6, 122, 3645) 36
834 1668 (12, 1666) 1666
835 1670 (12, 1668) 1668
836 1672 (12, 2, 5563) 556
837 1674 (12, 32, 1192, 3574) 357
838 1676 (12, 4, 20, 66, 1322, 6602) 660
839 1678 (12, 2, 123, 149, 8418) 84
840 1680 (12, 98, 112, 9916) 99
841 1682 (12, 202, 8202) 820
842 1684 (12, 2, 6, 86, 103, 244, 302, 4012, 1208) 120
843 1686 (12, 4, 2116, 8416) 84
844 1688 (12, 32, 2470) 24
845 1690 (12, 2, 5623) 562
846 1692 (12, 118, 18, 1988) 198
847 1694 (12, 1692) 1692
848 1696 (12, 2, 43, 2860) 28
849 1698 (12, 8482) 848
850 1700 (12, 5663) 566
851 1702 (12, 2, 32, 69, 187, 547, 1627) 162
852 1704 (12, 12, 130, 7802) 780
853 1706 (12, 4, 56, 1031, 2068) 20
854 1708 (12, 2, 2846) 284
855 1710 (12, 2447) 244
856 1712 (12, 28, 58, 8122) 812
857 1714 (12, 2, 11415) 114
858 1716 (12, 32, 4, 122, 212, 842, 1472, 5882) 588
859 1718 (12, 82, 100, 2008) 200
34
860 1720 (12, 2, 6, 952, 1902, 5702) 570
861 1722 (12, 2158) 215
862 1724 (12, 5743) 574
863 1726 (12, 2, 43, 112, 203, 222, 446, 2206) 220
864 1728 (12, 10, 523, 2606) 260
865 1730 (12, 32, 127, 187, 3642) 36
866 1732 (12, 2, 14412) 144
867 1734 (12, 1732) 1732
868 1736 (12, 4, 346, 6922) 692
869 1738 (12, 2, 6, 9618) 96
870 1740 (12, 232, 36, 8282) 828
871 1742 (12, 1740) 1740
872 1744 (12, 2, 32, 62, 823, 2466) 246
873 1746 (12, 4, 3485) 348
874 1748 (12, 1746) 1746
875 1750 (12, 2, 103, 523, 2606) 260
876 1752 (12, 82, 512, 4084) 408
877 1754 (12, 14612) 146
878 1756 (12, 2, 43, 6, 1247, 18, 3632) 36
879 1758 (12, 32, 505, 15010) 150
880 1760 (12, 8792) 879
881 1762 (12, 2, 5863) 586
882 1764 (12, 143, 202, 14012) 140
883 1766 (12, 4, 8820) 88
884 1768 (12, 2, 56, 106, 183, 9018) 90
885 1770 (12, 28, 60, 4204) 420
886 1772 (12, 32, 10, 112, 302, 334, 1102, 3304) 330
887 1774 (12, 2, 6, 1963, 5882) 588
888 1776 (12, 4, 20, 352, 14012) 140
889 1778 (12, 7424) 74
890 1780 (12, 2, 14812) 148
891 1782 (12, 12, 682, 2048) 204
892 1784 (12, 8912) 891
35
893 1786 (12, 2, 32, 43, 62, 830, 126, 2460) 24
894 1788 (12, 1786) 1786
895 1790 (12, 5963) 596
896 1792 (12, 2, 6, 992, 1988) 198
897 1794 (12, 10, 162, 8102) 810
898 1796 (12, 4, 1792, 7162) 716
899 1798 (12, 2, 2992, 5982) 598
900 1800 (12, 32, 1616, 4832) 48
901 1802 (12, 2572) 25
902 1804 (12, 2, 2524, 5024) 50
903 1806 (12, 4, 18, 362, 342, 6842) 684
904 1808 (12, 12, 138, 2766) 276
905 1810 (12, 2, 6, 18, 669, 1986) 198
906 1812 (12, 3625) 362
907 1814 (12, 32, 212, 367, 2526) 252
908 1816 (12, 2, 43, 103, 206, 1103, 2206) 220
909 1818 (12, 112, 392, 4294) 429
910 1820 (12, 82, 106, 4244) 424
911 1822 (12, 2, 3032, 6062) 606
912 1824 (12, 9112) 911
913 1826 (12, 4, 98, 20, 368, 1808) 180
914 1828 (12, 2, 32, 69, 283, 8420) 84
915 1830 (12, 56, 58, 2906) 290
916 1832 (12, 3056) 305
917 1834 (12, 2, 123, 232, 462, 2766) 276
918 1836 (12, 4, 1832, 7322) 732
919 1838 (12, 10, 832, 8302) 830
920 1840 (12, 2, 6123) 612
921 1842 (12, 32, 1312, 3934) 393
922 1844 (12, 18, 482, 14412) 144
923 1846 (12, 2, 43, 6, 122, 2030, 6020) 60
924 1848 (12, 9232) 923
925 1850 (12, 143, 6023) 602
36
926 1852 (12, 2, 15412) 154
927 1854 (12, 82, 363, 7224) 72
928 1856 (12, 32, 4, 122, 525, 15610) 156
929 1858 (12, 2, 6183) 618
930 1860 (12, 10, 12, 602, 156, 7802) 780
931 1862 (12, 1860) 1860
932 1864 (12, 2, 6, 112, 18, 222, 54, 662, 1982, 5942) 594
933 1866 (12, 4, 3725) 372
934 1868 (12, 1866) 1866
935 1870 (12, 2, 32, 62, 118, 228, 3316, 6616) 66
936 1872 (12, 9352) 935
937 1874 (12, 9362) 936
938 1876 (12, 2, 43, 203, 1003, 5003) 500
939 1878 (12, 1876) 1876
940 1880 (12, 9392) 939
941 1882 (12, 2, 6, 103, 189, 302, 9018) 90
942 1884 (12, 32, 268, 8042) 804
943 1886 (12, 4, 125, 285, 8420) 84
944 1888 (12, 2, 86, 363, 7224) 72
945 1890 (12, 4724) 472
946 1892 (12, 56, 6031) 60
947 1894 (12, 2, 4514, 9014) 90
948 1896 (12, 4, 378, 7562) 756
949 1898 (12, 32, 13514) 135
950 1900 (12, 2, 6, 2109) 210
951 1902 (12, 1900) 1900
952 1904 (12, 10, 172, 8602) 860
953 1906 (12, 2, 43, 718, 1418, 2854) 28
954 1908 (12, 1906) 1906
955 1910 (12, 112, 82, 9022) 902
956 1912 (12, 2, 32, 62, 1221, 212, 422, 8418) 84
957 1914 (12, 2398) 239
958 1916 (12, 4, 1912, 7642) 764
37
959 1918 (12, 2, 6, 18, 352, 702, 2102, 6302) 630
960 1920 (12, 18, 100, 9002) 900
961 1922 (12, 82, 284, 5632) 56
962 1924 (12, 2, 6430) 64
963 1926 (12, 32, 4, 10, 122, 2013, 302, 6026) 60
964 1928 (12, 202, 232, 4604) 460
965 1930 (12, 2, 2149) 214
966 1932 (12, 1930) 1930
967 1934 (12, 6443) 644
968 1936 (12, 2, 43, 6, 122, 149, 2818, 426, 8412) 84
969 1938 (12, 12, 148, 4444) 444
970 1940 (12, 32, 923, 2766) 276
971 1942 (12, 2, 3232, 6462) 646
972 1944 (12, 28, 66, 9242) 924
973 1946 (12, 4, 3885) 388
974 1948 (12, 2, 103, 583, 2906) 290
975 1950 (12, 1948) 1948
976 1952 (12, 9752) 975
977 1954 (12, 2, 32, 56, 69, 106, 1512, 3054) 30
978 1956 (12, 4, 810, 112, 442, 8820) 88
979 1958 (12, 18, 512, 3066) 306
980 1960 (12, 2, 6523) 652
981 1962 (12, 36, 52, 4684) 468
982 1964 (12, 12, 1510, 6030) 60
983 1966 (12, 2, 43, 1303, 2606) 260
984 1968 (12, 32, 704, 2108) 210
985 1970 (12, 10, 178, 8902) 890
986 1972 (12, 2, 6, 98, 18105) 18
987 1974 (12, 1972) 1972
988 1976 (12, 4, 20, 392, 1562, 7802) 780
989 1978 (12, 2, 6583) 658
990 1980 (12, 1978) 1978
991 1982 (12, 32, 943, 2826) 282
38
992 1984 (12, 2, 6603) 660
993 1986 (12, 4, 4445) 44
994 1988 (12, 1986) 1986
995 1990 (12, 2, 6, 86, 129, 2476) 24
996 1992 (12, 10, 18011) 180
997 1994 (12, 9962) 996
998 1996 (12, 2, 32, 43, 62, 126, 1821, 3642) 36
999 1998 (12, 1996) 1996
1000 2000 (12, 3336) 333
References
[1] M. Liebeck, A concise introduction to pure mathematicsThird edition (2010), CRC Press
[2] MATH08004, Group Theory: an Introduction to Abstract MathematicsMain Exam (2008), University of Edinburgh
[3] MATH08004, Group Theory: an Introduction to Abstract MathematicsMain Exam (2011), University of Edinburgh
39