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Natural Numbers God has created the Natural Numbers, but everything else is a mans work. Leopold Kronecker Leopold Kronecker Created by Inna Shapiro ©2006

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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.

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

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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 Peters number by 75. What was the original number that Peter wrote?

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Answer 91 91 Turn 91 upside-down, and you get Turn 91 upside-down, and you get 91-75 = 16

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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

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

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Problem 3 B) Construct 1,000,000 using six numbers 10. A) Can you get 1,000,000 using three numbers 100 and +, -, *

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

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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. ?

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Answer Turn 686 upside down, and you get 989.

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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?

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

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Problem 6 Mary wrote down all two-digit numbers. How many of them have at least one digit equal to 3?

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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.

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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?

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

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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?

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Answer 89,406 13,406

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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?

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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 is 314 – 142 = 172 pages

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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?

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

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