3.2 Prime numbers and prime factorisation — Practise with help

Warm-up: remember

Write down all the factors of 30.
Is 45 divisible by 3? Give a reason.

Answers and feedback

$1, 2, 3, 5, 6, 10, 15$ and $30$ . Working in pairs from 1 keeps you from missing one.
Yes. The digit sum is $4+5=9$ , and 9 is a multiple of 3. So $45\div3=15$ exactly.

Both answers are used in this section: the first tells you which numbers are composite, and the second is the kind of test you use to find a prime factor.

Practise with help

1. Which of these are prime numbers? $15, 17, 21, 27, 31$

Hint

Try dividing each one by 2, 3, 5 and 7 — the primes up to $\sqrt{31}$ , which is about 5.6.

Next step

15 and 27 divide by 3. 21 divides by 3 and by 7.

Full working

17 and 31 are prime.

$15=3\times5$ — composite
$17$ — prime
$21=3\times7$ — composite
$27=3\times3\times3$ — composite
$31$ — prime

To check 31 you need only try 2, 3 and 5. Each leaves a remainder, and $7^2=49$ is bigger than 31, so there is nothing further to try.

2. Write 60 as a product of its prime factors, in index form.

Hint

Divide by 2 while the number stays even, then move to the next prime.

Next step

$60\div2=30$ , $30\div2=15$ . Then 15 is odd, so divide by 3.

Full working

60 = 2\times2\times3\times5 = 2^2\times3\times5

Check by multiplying back: $4\times3\times5=60$ . Correct.

3. Write 84 as a product of its prime factors.

Hint

84 is even, so start with 2. Keep going while it stays even.

Next step

$84\div2=42$ , $42\div2=21$ . Now 21 is odd, and its digit sum is 3, so divide by 3.

Full working

84 = 2\times2\times3\times7 = 2^2\times3\times7

Check: $4\times3\times7=84$ . Correct.

4. Write 5775 as a product of its prime factors. It ends in 5, and its digit sum is 24.

Hint

Ending in 5 tells you 5 divides it; the digit sum being 24 tells you 3 divides it.

Next step

$5775\div3=1925$ . Then divide by 5: $1925\div5=385$ , and $385\div5=77$ .

Full working

5775 = 3\times5\times5\times7\times11 = 3\times5^2\times7\times11

Check: $3\times25=75$ , and $75\times7=525$ , and $525\times11=5775$ . Correct.

5. Write 1000 as a product of its prime factors, in index form.

Hint

1000 is $10\times10\times10$ . What is each 10 made of?

Next step

Each 10 is $2\times5$ . So 1000 is three 2s and three 5s.

Full working

1000 = 2^3\times5^3

Check: $8\times125=1000$ . Correct.

This is a useful one to recognise — 1000, 100 and 10 all split into equal numbers of 2s and 5s, which is exactly why the divisibility rules for 10, 100 and 1000 work.

6. $72 = 2^3\times3^2$ . Use the factorisation to find how many factors 72 has.

Hint

A factor uses some number of 2s and some number of 3s. How many choices for each?

Next step

0, 1, 2 or 3 copies of 2 — that is 4 choices. And 0, 1 or 2 copies of 3 — 3 choices.

Full working

(3+1)\times(2+1) = 4\times3 = 12

So 72 has 12 factors. You can check the count by listing them: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36$ and $72$ .

7. Explain why 1 is not a prime number.

Hint

Count the factors of 1. How many are there?

Next step

1 has exactly one factor.

Full working

A prime number has exactly two factors: 1 and itself. The number 1 has only one factor, because "1 and itself" is the same number counted once.

So 1 fails the definition. This also matters for factorisation: if 1 were allowed as a prime, you could write $12 = 1\times1\times2\times2\times3$ using as many 1s as you liked, and the factorisation would no longer be unique.

8. A number's prime factorisation is $2\times3^3\times5^2$ . Without working out the number, say how many factors it has.

Hint

Use the same rule: add one to each index and multiply.

Next step

$(1+1)\times(3+1)\times(2+1)$ .

Full working

2\times4\times3 = 24

So it has 24 factors. You did not need to know that the number is 1350 — the factorisation alone answers the question.

A question from an earlier section

9. Write down all the factors of 40.

Hint

Work in pairs from 1.

Next step

$40=1\times40=2\times20=4\times10=5\times8$ .

Full working

$1, 2, 4, 5, 8, 10, 20$ and $40$ .

Now check it with this section's method. $40=2^3\times5$ , so the number of factors is $(3+1)\times(1+1)=8$ . There are eight in the list above, which agrees.

That is a useful check: counting with the formula tells you how many factors to look for, so you know when your list is complete.