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EXAMPLE 123452• Since 52=13(4) is divisible by 4, 123452 is divisible by 4• Since 452=56(8)+4 is not divisible by 8, 123452 is not divisible by 8• 123452 - twice the last digit is 2(2)=4 and

12345-4=12341• 12341 – twice the last digit is 2(1)=2 and

1234-2=1232• 1232 – twice the last digit is 2(2) =4 and

123-4=119• 119 – twice the last digit is 2(9)=18 and 11-18=-7 is divisible by 7

123452 IS DIVISIBLE BY 7

EXAMPLE IS 1234567 DIVISIBLE BY 13?

• USE THE LIST 1,10,9,12,3,4 REPEATEDLY AS NEEDED1x7 + 10x6 + 9x5 + 12x4 + 3x3 + 4x2 + (start over) + 1x1 = 7+60+45+48+9+8+1=67+93+17+1=160+18=178NOW DO IT AGAIN!1x8 + 10x7 + 9x1 = 8+70+9=70+17=87 NOW DO IT AGAIN! (DOESN’T HELP)1x7+10x8=7+80=87Since 87=6(13)+9 The remainder when dividing 87 by 13 is 9 and so the remainder when dividing 1234567 by 13 is also 9.

The Euclidean AlgorithmTo Find Greatest Common Divisors

WITHOUT FACTORING!

IF YOU WANT TO FIND GCD(a,b) then Note that if any number D divides a and b then it will also divide

a-Nbfor any positive integer N. So this means that

GCD(a,b)=GCD(b,a-Nb)We make these numbers smaller and continue the thinking!

SO NOTE THE FOLLOWING!IN EVERY SLIDE WE WILL ASK A QUESTION WITH THE SAME ANSWER! THE QUESTIONS ARE GETTING EASIER AND EASIER!

THIS IS COMMON IN MATHEMATICS – FOR EXAMPLE WE MIGHT ASK HOW DO YOU SOLVE 2X+3=19? WE CHANGE THE QUESTION TOHOW DO YOU SOLVE 2X=16?THEN HOW DO YOU SOLVE X=8?I.E. YOU KEEP GOING UNTIL THE ANSWER IS OBVIOUS!

FIND GREATEST COMMON DIVISOR OF 15158 AND 6307

WHAT IS THE QUOTIENT AND REMAINDER WHEN DIVIDING 15158 BY 6307?

15158=2(6307)+2544

NOW FIND GREATEST COMMON DIVISOR OF 6307 AND 2544

WHAT IS THE QUOTIENT AND REMAINDER WHEN DIVIDING 6307 BY 2544?

6307=2(2544)+1219

NOW FIND GREATEST COMMON DIVISOR OF 2544 AND 1219

WHAT IS THE QUOTIENT AND REMAINDER WHEN DIVIDING 2544 BY 1219?

2544=2(1219)+106

NOW FIND GREATEST COMMON DIVISOR OF 1219 AND 106

WHAT IS THE QUOTIENT AND REMAINDER WHEN DIVIDING 1219 BY 106?

1219=11(106)+53

NOW FIND GREATEST COMMON DIVISOR OF 106 AND 53

WHAT IS THE QUOTIENT AND REMAINDER WHEN DIVIDING 106 BY 53?

106=2(53)+0The zero says we are done so

GCD(15158,6307)=53

EXAMPLES FOR YOU TO TRY!

FIND THE GCD OF 23 and 123.

FIND the GCD of 12356 and 12346.

FIND the GCD of TWO SOCIAL SECURITY NUMBERS.

Divisibility Puzzle• Form an integer by using each of {1,2,3,4,5,6,7,8,9}Exactly once so that• The first k-digits are divisible by k for k=1,2,…,9

Divisibility Puzzle EXAMPLEThe first k-digits are divisible by kFor Example 123456789

Partially works since 1 is divisible by 1 12 is divisible by 2

123 is divisible by 3BUT 1234 is not divisible by 4.

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