Natural NumbersNatural Numbers

”God has created the Natural Numbers, but everything else is a man’s work.”

LeopoldLeopold KroneckerKronecker

Created by Inna Shapiro ©2006

What are natural numbers?

Natural numbers are

the ones that we use to

count: 1, 2, 3,4,5… etc. For example, all of us

know that 4 quarters

make a dollar, but in

Math we write it as

4*25 = 100

.

In this section we will explore the magical relations between the natural numbers.

What are natural numbers?

Problem 1 Peter wrote in notepad on his

desk a two-digit number. Adam, who sat right in front of Peter, turned back and saw another number, that was actually less than Peter’s number by 75.

What was the original number that Peter wrote?

Answer

9191 Turn 91 upside-down, and you Turn 91 upside-down, and you

getget

91-75 = 16 91-75 = 16

Problem 2

Write twenty 5-s. Place some addition

signs between them, so that

the result would be 1000

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Answer

555+55+55+55+55+55+55+55+55+5=1.000

Problem 3

B) Construct 1,000,000 using six numbers 10.

A) Can you get 1,000,000 using

three numbers 100 and “ + ”, “ - ”, “ * ”

Answer

A) 1,000,000 = 100*100*100 B) 1,000,000 = 10*10*10*10*10*10

Problem 4

Adam wrote 686. How can he get a number

greater than this one by 303

without doing any addition /

subtraction / multiplication /

division etc. ?

Answer

Turn 686 upside down, and you get 989.

Problem 5

Can you write a seven-digit number so that

the sum of its digits is two? How many of such numbers are there? Can you write all such numbers?

Answer 2,000,000 1,100,000 1,010,000 1,001,000 1,000,100 1,000,010 1,000,001

Problem 6

Mary wrote down all two-digit numbers.

How many of them have at least one digit equal to 3?

Answer

Ten numbers begin with 3: 30, 31, 32, 33, 34, 35, 36, 37, 38, 39

and nine numbers end with 3:3, 13, 23, 33, 43, 53, 63, 73, 83, 93,

but we counted 33 twice, so there are 18 numbers that contain at least one digit equal to 3.

Problem 7

Helen tried to use every digit to compose a number. What is the

smallest number she can compose?

What is the smallest such number

that could be divided by 5?

Answer

1,023,456,789 1,023,467,895

Problem 8

Bill wrote number 513,879,406 and asked his Dad to erase four digits in two different ways in order to get the maximal and minimal answers possible. Which four digits should his dad remove in both cases and what numbers is he going to get?

Answer

89,406 13,406

Problem 9

A block of pages is missing from a book. The number of the first missing page is 143, and the number of the last one is made of the same three digits.

How many pages are missing?

Answer

The number of last missing page could

be 341,431,413 or 314. It has to be even, because the first one was odd. So it has to be 314, and the total number of pages missing is314 – 142 = 172 pages

Problem 10

How many four-digit numbers can you

write using the digits 0, 1, 2 and 3, in such a way that 0 and 2 are not adjacent?

Answer

8 numbers: 2310, 2130, 2301, 2103, 3210, 3012, 1230, 1032.