Prime Numbers Created by Inna Shapiro 2008 Definition

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.

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?

Answer The youngest kid is 5 years old. The rest are 7, 11, 13, 17 and 19 years old.

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?

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

Problem 3 Is the following number prime? 2001 + 2007 ?

Answer The last digit of 2001 is 1. The last digit of 2007 is an odd number, because the product of any number of odd integers is odd. That means 2001 + 2007 is even and cannot be a prime number.

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

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).

Problem 5 A teacher wrote nine numbers on the blackboard: 123456789 and asked the students to put “+” and “-” signs between them to get as many two-digit prime numbers as possible. Can you do it?

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: 1 + 2 + 3 + 4 + 5 + 6 + 7 - 8 – 9 = 11 1 + 2 + 3 + 4 + 5 + 6 – 7 + 8 – 9 = 13 1 + 2 + 3 + 4 - 5 + 6 + 7 + 8 – 9 = 17

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.

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.

Problem 7 Max has two cards with prime numbers A and B. He said that the last digit of the sum A 2+B 2 is 9. Can you find A and B?

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 A 2=4 and the last digit of B 2 is 5. That means B = 5, because B is divisible by 5. So A = 2, B = 5 and A 2+B 2 = 29.

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.

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.

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

Answer Let us try to divide A by 3. The residual can be 0, 1, or 2. That means A could be written as: 1. A = 3 * k for some integer k, or 2. A = 3 * k + 1 for some integer k, or 3. 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.

Problem 10 There are three consecutive odd prime numbers 3, 5, and 7. Are there any other three consecutive odd prime numbers?

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.