Created by Inna Shapiro ©2008

Prime Prime NumbersNumbers

A prime number is an integer A prime number is an integer greater than 1 that has exactly greater than 1 that has exactly two divisors, 1 and itself.two divisors, 1 and itself.

The first ten prime numbers are The first ten prime numbers are

2, 3, 5, 7, 11, 13, 17, 19, 23, and 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.29.

Integers that are not prime are Integers that are not prime are called composite numbers.called composite numbers.

DefinitionDefinition

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

Problem 1Problem 1

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

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

AnswerAnswer

Mary wrote four consecutive Mary wrote four consecutive prime numbers. Then she prime numbers. Then she calculated their product and calculated their product and got a number whose last got a number whose last digit is 0.digit is 0.

What numbers did she write?What numbers did she write?

What was the product?What was the product?

Problem 2Problem 2

The product is divisible by 10, The product is divisible by 10, that means two of the factors that means two of the factors were 2 and 5, because no were 2 and 5, because no other prime number can be other prime number can be divisible by 2 or 5.divisible by 2 or 5.

We can conclude that Mary We can conclude that Mary wrote 2, 3, 5, and 7.wrote 2, 3, 5, and 7.

AnswerAnswer

The product is The product is

2 * 3 * 5 * 7 = 2102 * 3 * 5 * 7 = 210

Is the following number prime?Is the following number prime?

2001200120012001 + 2007 + 20072007 2007 ??

Problem 3Problem 3

The last digit of 2001The last digit of 20012001 2001 is 1.is 1.

The last digit of 2007The last digit of 20072007 2007 is an odd is an odd number, because the product of number, because the product of any number of odd integers is odd.any number of odd integers is odd.

That means 2001That means 200120012001 + 2007 + 200720072007 is is even and cannot be a prime even and cannot be a prime number.number.

AnswerAnswer

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

Problem 4Problem 4

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

is 45, and it is divisible by 3.is 45, and it is divisible by 3.

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

AnswerAnswer

A teacher wrote nine numbers A teacher wrote nine numbers on the blackboard:on the blackboard:

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

and asked the students to put and asked the students to put “+” and “-” signs between “+” and “-” signs between them to get as many two-them to get as many two-digit prime numbers as digit prime numbers as possible.possible.

Can you do it?Can you do it?

Problem 5Problem 5

We can never get a number bigger We can never get a number bigger than 45, because 1+2+…+9 = than 45, because 1+2+…+9 = 45.45.

There are ten two-digit prime There are ten two-digit prime numbers less than 45: numbers less than 45: 11,13,17,19,23,29,31,37,41,43. 11,13,17,19,23,29,31,37,41,43.

Look how we can get these Look how we can get these numbers:numbers:

1 + 2 + 3 + 4 + 5 + 6 + 7 - 8 – 9 = 1 + 2 + 3 + 4 + 5 + 6 + 7 - 8 – 9 = 1111

1 + 2 + 3 + 4 + 5 + 6 – 7 + 8 – 9 = 1 + 2 + 3 + 4 + 5 + 6 – 7 + 8 – 9 = 1313

1 + 2 + 3 + 4 - 5 + 6 + 7 + 8 – 9 = 1 + 2 + 3 + 4 - 5 + 6 + 7 + 8 – 9 = 1717

AnswerAnswer

1 + 2 + 3 - 4 + 5 + 6 + 7 + 8 – 9 = 1 + 2 + 3 - 4 + 5 + 6 + 7 + 8 – 9 = 1919

1 – 2 + 3 + 4 + 5 + 6 + 7 + 8 – 9 = 1 – 2 + 3 + 4 + 5 + 6 + 7 + 8 – 9 = 2323

1 + 2 + 3 + 4 + 5 + 6 + 7 - 8 + 9 = 1 + 2 + 3 + 4 + 5 + 6 + 7 - 8 + 9 = 2929

1 + 2 + 3 + 4 + 5 + 6 - 7 + 8 + 9 = 1 + 2 + 3 + 4 + 5 + 6 - 7 + 8 + 9 = 3131

1 + 2 + 3 – 4 + 5 + 6 + 7 + 8 + 9 = 1 + 2 + 3 – 4 + 5 + 6 + 7 + 8 + 9 = 3737

1 - 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 1 - 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 4141

2 – 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 2 – 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 4343

Answer Answer /continued//continued/

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

Problem 6Problem 6

Two-digit prime numbers could end Two-digit prime numbers could end only with 1, 3, 7, or 9. We get four only with 1, 3, 7, or 9. We get four pairs of two-digit prime numbers, pairs of two-digit prime numbers, which could be written with the same which could be written with the same digits:digits:

31 and 13, 31 – 13 = 18;31 and 13, 31 – 13 = 18;

71 and 17, 71 – 17 = 54;71 and 17, 71 – 17 = 54;

97 and 79, 97 – 79 = 18;97 and 79, 97 – 79 = 18;

73 and 37, 73 – 37 = 36, where 36 = 673 and 37, 73 – 37 = 36, where 36 = 622..

The answer is 37 and 73.The answer is 37 and 73.

AnswerAnswer

Max has two cards with prime Max has two cards with prime numbers A and B. He said numbers A and B. He said that the last digit of the sum that the last digit of the sum AA22+B+B22 is 9. is 9.

Can you find A and B?Can you find A and B?

Problem 7Problem 7

If a sum of two numbers ends with 9, If a sum of two numbers ends with 9, then one number is even, and the then one number is even, and the other is odd. An even number cannot other is odd. An even number cannot be a square of any prime number be a square of any prime number other than 2. That means that either other than 2. That means that either A or B is 2. Suppose A = 2, then AA or B is 2. Suppose A = 2, then A22=4 =4 and the last digit of Band the last digit of B2 2 is 5. That is 5. That means B = 5, because B is divisible by means B = 5, because B is divisible by 5. 5.

So A = 2, B = 5 and ASo A = 2, B = 5 and A22+B+B22 = 29. = 29.

AnswerAnswer

Ann has three cards with Ann has three cards with different digits. She stated different digits. She stated that she can compose six that she can compose six different three-digit prime different three-digit prime numbers using these cards.numbers using these cards.

Prove that Ann is wrong.Prove that Ann is wrong.

Problem 8Problem 8

All digits must be odd, because a three-digit All digits must be odd, because a three-digit number with an even last digit is divisible number with an even last digit is divisible by 2.by 2.

There is no 5 on her cards, because a three-There is no 5 on her cards, because a three-digit number with last digit 5 is divisible by digit number with last digit 5 is divisible by 5.5.

So the digits on Ann’s cards can only be 1, 3, So the digits on Ann’s cards can only be 1, 3, 7, or 9.7, or 9.

There are only 6 ways to arrange 3 cards, so There are only 6 ways to arrange 3 cards, so any arrangement of the cards must give a any arrangement of the cards must give a prime number.prime number.

If she has 1,3,7, then 371 = 53 * 7;If she has 1,3,7, then 371 = 53 * 7;

If she has 1,3,9, then 319 = 29 * 11;If she has 1,3,9, then 319 = 29 * 11;

If she has 1,7,9, then 791 = 113 * 7;If she has 1,7,9, then 791 = 113 * 7;

If she has 3,7,9, then 793 = 61 * 13.If she has 3,7,9, then 793 = 61 * 13.

That means Ann made a mistake and there is That means Ann made a mistake and there is no such set of cards.no such set of cards.

AnswerAnswer

Can you find a prime number A Can you find a prime number A so that (A + 10) and (A + 14) so that (A + 10) and (A + 14) are also prime numbers?are also prime numbers?

Find all possible answers.Find all possible answers.

Problem 9Problem 9

Let us try to divide A by 3. The residual can be Let us try to divide A by 3. The residual can be 0, 1, or 2. That means A could be written as:0, 1, or 2. That means A could be written as:

1.1.A = 3 * k for some integer k, orA = 3 * k for some integer k, or

2.2.A = 3 * k + 1 for some integer k, orA = 3 * k + 1 for some integer k, or

3.3.A=3 * k + 2 for an integer kA=3 * k + 2 for an integer k

If A = 3 * k, A is prime only if k = 1, thenIf A = 3 * k, A is prime only if k = 1, then

A + 10 = 13, and A + 14 = 17.A + 10 = 13, and A + 14 = 17.

All three numbers 3,13, and 17 are prime All three numbers 3,13, and 17 are prime numbers.numbers.

If A = 3 * k + 1, then If A = 3 * k + 1, then

A + 14 = 3 * k + 15 => divisible by 3.A + 14 = 3 * k + 15 => divisible by 3.

If A = 3 * k + 2, then If A = 3 * k + 2, then

A + 10 = 3 * k + 12 => divisible by 3.A + 10 = 3 * k + 12 => divisible by 3.

We see that the only possible answer is A = 3.We see that the only possible answer is A = 3.

AnswerAnswer

There are three consecutive There are three consecutive odd prime numbers 3, 5, and odd prime numbers 3, 5, and 7.7.

Are there any other three Are there any other three consecutive odd prime consecutive odd prime numbers? numbers?

Problem 10Problem 10

No, there are not.No, there are not.

Suppose we have three consecutive odd Suppose we have three consecutive odd numbers. We can write these numbers as A, numbers. We can write these numbers as A, A + 2, and A + 4.A + 2, and A + 4.

A is not divisible by 3, otherwise A would not A is not divisible by 3, otherwise A would not be prime. That means A can be written as be prime. That means A can be written as either A=3*k+1 or A=3*k+2 for some either A=3*k+1 or A=3*k+2 for some integer k.integer k.

If A= 3*k+1, then A+2 is divisible by 3, so A+2 If A= 3*k+1, then A+2 is divisible by 3, so A+2 is not prime. is not prime.

If A= 3*k+2, then A+4 is divisible by 3, which If A= 3*k+2, then A+4 is divisible by 3, which means A+4 is not prime.means A+4 is not prime.

AnswerAnswer