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**Created by Inna Shapiro ©2008**

Prime Numbers Created by Inna Shapiro ©2008 1

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**The first ten prime numbers are **

Definition A prime number is an integer greater than 1 that has exactly two divisors, 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Integers that are not prime are called composite numbers. 2

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Problem 1 There are six children in a family. Five of them are older than the youngest one by 2,6,8,12 and 14 years. How old are they if the age of every kid is a prime number? 3

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**Answer The youngest kid is 5 years old.**

The rest are 7, 11, 13, 17 and 19 years old. 4

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Problem 2 Mary wrote four consecutive prime numbers. Then she calculated their product and got a number whose last digit is 0. What numbers did she write? What was the product? 5

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Answer The product is divisible by 10, that means two of the factors were 2 and 5, because no other prime number can be divisible by 2 or 5. We can conclude that Mary wrote 2, 3, 5, and 7. The product is 2 * 3 * 5 * 7 = 210 6

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Problem 3 Is the following number prime? ? 7

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**Answer The last digit of 20012001 is 1.**

The last digit of is an odd number, because the product of any number of odd integers is odd. That means is even and cannot be a prime number. 8

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Problem 4 Dan has nine cards with the digits 1,2,…9. He arranged these cards in a random order to compose a nine-digit number. Is that number prime or composite? 9

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**Answer The sum of the nine digits 1, 2, … 9**

is 45, and it is divisible by 3. So Dan will always get a composite number (a number is divisible by 3 if the sum of its digits is divisible by 3). 10

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**Problem 5 A teacher wrote nine numbers on the blackboard:**

and asked the students to put “+” and “-” signs between them to get as many two-digit prime numbers as possible. Can you do it? 11

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**Answer We can never get a number bigger than 45, because 1+2+…+9 = 45.**

There are ten two-digit prime numbers less than 45: 11,13,17,19,23,29,31,37,41,43. Look how we can get these numbers: – 9 = 11 – – 9 = 13 – 9 = 17 12

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**Answer /continued/ 1 + 2 + 3 - 4 + 5 + 6 + 7 + 8 – 9 = 19**

1 – – 9 = 23 = 29 = 31 – = 37 = 41 2 – = 43 13

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Problem 6 Please find two different two-digit prime numbers such that when you write one of them backwards, you get the other, and the difference between these numbers is a perfect square. 14

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Answer Two-digit prime numbers could end only with 1, 3, 7, or 9. We get four pairs of two-digit prime numbers, which could be written with the same digits: 31 and 13, 31 – 13 = 18; 71 and 17, 71 – 17 = 54; 97 and 79, 97 – 79 = 18; 73 and 37, 73 – 37 = 36, where 36 = 62. The answer is 37 and 73. 15

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Problem 7 Max has two cards with prime numbers A and B. He said that the last digit of the sum A2+B2 is 9. Can you find A and B? 16

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Answer If a sum of two numbers ends with 9, then one number is even, and the other is odd. An even number cannot be a square of any prime number other than 2. That means that either A or B is 2. Suppose A = 2, then A2=4 and the last digit of B2 is 5. That means B = 5, because B is divisible by 5. So A = 2, B = 5 and A2+B2 = 29. 17

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Problem 8 Ann has three cards with different digits. She stated that she can compose six different three-digit prime numbers using these cards. Prove that Ann is wrong. 18

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Answer All digits must be odd, because a three-digit number with an even last digit is divisible by 2. There is no 5 on her cards, because a three-digit number with last digit 5 is divisible by 5. So the digits on Ann’s cards can only be 1, 3, 7, or 9. There are only 6 ways to arrange 3 cards, so any arrangement of the cards must give a prime number. If she has 1,3,7, then 371 = 53 * 7; If she has 1,3,9, then 319 = 29 * 11; If she has 1,7,9, then 791 = 113 * 7; If she has 3,7,9, then 793 = 61 * 13. That means Ann made a mistake and there is no such set of cards. 19

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Problem 9 Can you find a prime number A so that (A + 10) and (A + 14) are also prime numbers? Find all possible answers. 20

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Answer Let us try to divide A by 3. The residual can be 0, 1, or 2. That means A could be written as: A = 3 * k for some integer k, or A = 3 * k + 1 for some integer k, or A=3 * k + 2 for an integer k If A = 3 * k, A is prime only if k = 1, then A + 10 = 13, and A + 14 = 17. All three numbers 3,13, and 17 are prime numbers. If A = 3 * k + 1, then A + 14 = 3 * k + 15 => divisible by 3. If A = 3 * k + 2, then A + 10 = 3 * k + 12 => divisible by 3. We see that the only possible answer is A = 3. 21

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**Problem 10 There are three consecutive odd prime numbers 3, 5, and 7.**

Are there any other three consecutive odd prime numbers? 22

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Answer No, there are not. Suppose we have three consecutive odd numbers. We can write these numbers as A, A + 2, and A + 4. A is not divisible by 3, otherwise A would not be prime. That means A can be written as either A=3*k+1 or A=3*k+2 for some integer k. If A= 3*k+1, then A+2 is divisible by 3, so A+2 is not prime. If A= 3*k+2, then A+4 is divisible by 3, which means A+4 is not prime. 23

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