**Week 4: Prime Numbers – Day 5**

1. The prime decomposition of a number is 2 x 11. What are all possible factors of the number?

The possible factors are 1, 2, 11, and 22.

2. The prime decomposition of a number is 2 x 3 x 3 x 5. What are all possible factors of the number?

The factors are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

3. The prime decomposition of a number is 2 x 3 x 5 x 7. Is this number divisible by 21?

Yes, because 3 x 7 is a factor and 3 x 7 is 21.

4. The prime decomposition of a number is 2 x 2 x 5 x 5 x 5. How many zeros are at the right end of the number (in other words, how many multiples of 10 can go into the number)?

10 = 2 x 5. The number can be rearranged as 2 x 5 x 2 x 5 x 5. Therefore the number is

10 x 10 x 5 = 500. There are two zeros at the end of the number.

5. Sally has opens her book. The product of the two open page numbers is 210. What are the page numbers?

The prime factorization of 210 is 2 x 3 x 5 x 7. Here are all its factors:

1 and 210

2 and 105

3 and 70

5 and 42 (because 42 = 2 x 3 x 7)

6 and 35 (because 6 = 2 x 3, and 35 = 5 x 7)

— and so on. There are two factors which are consecutive numbers: 14 and 15. (2 x 7 and 3 x 5). These must be the open page numbers.

**Week 4: Prime Numbers – Day 4**

1. What are the prime factors of 6?

6 = 2 x 3 – the factors are 2, 3

2. What are the prime factors of 42?

42 = 2 x 3 x 7 – the factors are 2, 3, 7

3. What are the prime factors of 32?

32 = 2 x 2 x 2 x 2 x 2

32 has only one prime factor: 2

4. Write the number 45 as a product of primes.

45 = 3 x 3 x 5

5. Write the number 84 as a product of primes.

84 = 2 x 2 x 3 x 7

**Week 4: Prime Numbers – Day 3**

How Many Prime Numbers Are There?

1. How many prime numbers are less than 10?

Let’s count them: they are 2, 3, 5, and 7. Four prime numbers less than 10.

2. How many prime numbers are between 10 and 20?

The prime numbers between 10 and 20 are: 11, 13, 17 and 19. Four prime numbers between 10 and 20.

3. How many prime numbers are between 20 and 30?

The prime numbers between 20 and 30 are: 23 and 29. Two prime numbers between 20 and 30.

4. How many prime numbers are between 30 and 40?

The prime numbers between 30 and 40 are: 31 and 37. Two prime numbers between 30 and 40.

(note there are three prime numbers between 40 and 50. What are they?)

5. In a group of ten consecutive numbers > 10, what is the maximum number of primes there could be? (Hint: how many of the ten numbers MUST be even? Knock those out. What other multiples can you eliminate?) It might help to write the numbers down as a list and cross out the possibilities. This is a long problem to answer fully. Give yourself full credit if you narrow down the possibilities to five.

Let the ten consecutive numbers be A, B, C, D, E, F, G, H, I, and J.

If A is even, so are C, E, G, and I. The five numbers A, C, E, G, and I cannot be prime. We will write our list of non-primes as: *A, B, *C, D, *E, F, *G, H, *I, J.

One of the numbers A, B, or C is divisible by 3. (Try it out: of any three consecutive numbers, one MUST be a multiple of 3. Can you explain why?)

If A is divisible by 3, then so is D. If D is divisible by 3, then so is G. And so is J. Now if we put * beside numbers that are composite, our list looks like this:

*A, B, *C, *D, *E, F, *G, H, *I, *J – so if A even and is divisible by 3, there are at most 2 primes.

If, instead of A, B is divisible by 3, then so is E and so is H. The list looks like:

*A, *B, *C, D, *E, F, *G, *H, *I, J – so if A is even and B is divisible by 3, there are at most 3 primes.

If, instead of B, C is divisible by 3, so are F and I. the list becomes:

*A, B, *C, D, *E, *F, *G, H, *I, J – so if A is even and C is divisible by 3, there are at most 3 primes.

There will be at least one multiple of 5 in the set (up to two), and one multiple of 7. (Think about that). In some cases, the multiple of 5 will coincide with a multiple of 2 or 3, and in other cases it won’t. Similarly, the multiple of 7 will coincide or it won’t.

Here’s a way to think count things up:

There are always 5 multiples of 2. That leaves 5 possibilities out of 10.

There are always 3 multiples of 3. That might rule out 3 more possibilities, but, there is always at least one multiple of 6 (simultaneously a multiple of 2 and a multiple of 3). Therefore only 2 more possibilities are ruled out. That leaves 3 possibilities out of 10.

There is always at least one multiple of 5, but it might overlap a multiple of 2 or 3.

There is always at least one multiple of 7 in the set, but it might overlap a multiple of 2 or 3.

Therefore the maximum number of primes there could be is 3. (Between 50 and 60, for example, are 3 primes. What are they?)

**Week 4: Prime Numbers – Day 2**

1. Which of the following numbers is divisible by 2 (and definitely not prime)?

17, 26, 101, 924, 1000000, 39

The even numbers ending in 6, 4, and 0 are divisible by 2:

26, 924, and 1000000

2. Which of the following numbers is divisible by 3 (and definitely not prime)?

16, 51, 39, 243, 117, 2014, 2016

If a number is divisible by 3, the sum of its digits is divisible by 3.

The numbers divisible by 3 are: 51, 39, 243, 117, and 2016.

3. Which of the following numbers is divisible by 9 (and definitely not prime)?

16, 51, 39, 72, 243, 117, 2010, 2014, 2016

If a number is divisible by 9, the sum of its digits is divisible by 9.

The numbers divisible by 9 are: 72, 243, 117, and 2016.

4. Which of the following numbers is divisible by 7?

18, 28, 43, 45, 49, 52, 63, 72, 87, 91, 98

Using knowledge of multiples of 7, the answers are: 28, 49, 63, 91, and 98.

5. Which of the following numbers is divisible by 15?

16, 51, 39, 72, 243, 117, 2010, 2014, 2016

A number divisible by 15 has to be divisible by both 3 and 5.

The only number in the list that is divisible by 5 is 2010. It is also divisible by 3, since 2 + 0 + 1 + 0 is 3, and therefore the answer is 2010.

**Week 4: Prime Numbers – Day 1**

1. How many even prime numbers are there?

One. The number 2 is the only even prime number. It’s the smallest prime number. All of the bigger ones have to be odd, or else 2 would be a factor.

2. List all the prime numbers between 10 and 20.

11, 13, 17, 19

3. What is the name for a whole number > 1 that is not prime?

Composite number.

4. List three consecutive prime numbers.

There are many answers: 2, 3, 5, or 11, 13, 17, or 23, 29, 31. Check your answer with a friend.

5. Is it true that the sum of two prime numbers is a prime number?

No, because it is false for at least one example. 2 + 7 = 9, and 9 has factors 1, 3, and 9, so 9 is not prime. Also 2 + 2 = 4, and 4 is not prime either.