3.2 Prime numbers and prime factorisation — Practise

1. Write down all the prime numbers between 20 and 40.

Answer: $23, 29, 31$ and $37$. Each has no factor smaller than itself except 1: 21, 27 and 33 divide by 3, 25 and 35 by 5, 39 by 3, and 22, 24, 26, 28, 30, 32, 34, 36 and 38 are even.

2. Show that 91 is not a prime number.

Answer: $91=7\times13$, so 91 has a factor other than 1 and itself. It is not prime.

3. Write 360 as a product of its prime factors.

Answer: $360=2^3\times3^2\times5$. Divide out 2 three times, then 3 twice, then 5 once.

4. Write 700 as a product of its prime factors.

Answer: $700=2^2\times5^2\times7$.

5. How many factors does 84 have?

Answer: Twelve. Each factor uses 0, 1 or 2 copies of 2, and 0 or 1 copy of each of 3 and 7 — that is $3\times2\times2=12$ combinations, and each gives a different factor.

6. Write 5775 as a product of prime factors.

Answer: $5775=3\times5^2\times7\times11$.