Prime numbers and factorisation

A prime number has exactly two factors: 1 and itself. A composite number has more than two. Every number above 1 can be written as a product of primes in exactly one way, which is its prime factorisation.

Formulas

The primes below 30

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

A number is divisible by

2 (last digit even), 3 (digit sum divisible by 3), 5 (ends in 0 or 5)

Worked examples

Example 1. Is 91 a prime number?

Try small primes: 91 is not divisible by 2, 3 or 5.
$91 = 7 \times 13$ , so it has more factors than 1 and 91.

Answer: No, 91 is composite

Example 2. Write 180 as a product of primes.

$180 = 2 \times 90 = 2 \times 2 \times 45$ .
$45 = 3 \times 15 = 3 \times 3 \times 5$ .
$S o 180 = 2^{2} \times 3^{2} \times 5$ .

Answer:

2^{2} \times 3^{2} \times 5

Example 3. How many prime numbers are there between 20 and 40?

Check each number: 23, 29, 31 and 37 have no factors other than 1 and themselves.
21, 25, 27, 33, 35, 39 are divisible by 3, 5 or 7, and the even numbers are composite.

Answer: 4 primes

Common mistakes

Calling 1 a prime. It has only one factor, so it is neither prime nor composite.
Thinking all odd numbers are prime. 9, 15, 21, 27, 33, 35 and 91 are odd but composite.
Forgetting that 2 is prime. It is the only even prime.

Practice questions with answers

1. Which of the following is a prime number?

Show answer

Answer: B) 11

11 has only two factors, 1 and 11, so it is prime.

9 = 3 \times 3, 15 = 3 \times 5, 21 = 3 \times 7

are composite.

2. Find the cube root of 8 by prime factorisation.

Show answer

Answer: A) 2

8 =

2 \times 2 \times 2 = 2^{3}

. Grouping one triple of 2, cube root = 2.

3. Which of these numbers is prime?

Show answer

Answer: C) 29

21 =

3 \times 7, 27 = 3 \times 9

and

33 = 3 \times 11

all have other factors. 29 has only the factors 1 and 29, so it is prime.

4. Reena thinks of a two-digit number. When she reverses its digits she gets another number, and BOTH numbers are prime. For example, 13 and 31 are both prime.
Which of the following is another such pair?

A17 and 71
B21 and 12
C27 and 72
D39 and 93

Show answer

Answer: A) 17 and 71

17 and 71 are both prime, so reversing the digits keeps it prime. In the other pairs at least one number is composite

(21 = 3 \times 7, 27 = 3 \times 9, 39 = 3 \times 13

, etc.).

5. The number 137592 is not a perfect cube.
Its prime factorisation is

137592 = 2^{3} \times 3^{4} \times 7^{3}

.
Find the smallest number it must be divided by to give a perfect cube, and the cube root of the result.

Adivide by 3, cube root 42
Bdivide by 9, cube root 42
Cdivide by 3, cube root 14
Ddivide by 27, cube root 14

Show answer

Answer: A) divide by 3, cube root 42

137592 =

2^{3} \times 3^{4} \times 7^{3}

. The 3 has exponent 4; remove one 3 (divide by 3) to leave

3^{3}

. Result =

2^{3} \times 3^{3} \times 7^{3} = (2 x 3 x 7)^{3} = 4 2^{3}

, cube root = 42.

6. A plot in Jaipur has an area of 1960 square metres. Write 1960 in prime-power form.

A $2^{3} \times 5 \times 7^{2}$
B $2^{3} \times 5 \times 7$
C $2^{3} \times 5^{2} \times 7$
D $2^{2} \times 5 \times 7^{2}$

Show answer

Answer: A)

2^{3} \times 5 \times 7^{2}

1960 =

196 \times 10 = (2^{2} \times 7^{2}) \times (2 \times 5) = 2^{3} \times 5 \times 7^{2}

. Check:

8 \times 5 \times 49 = 1960

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Frequently asked questions

What is a prime number?

A number greater than 1 whose only factors are 1 and itself, such as 2, 3, 5, 7 and 11.

Is 1 a prime number?

No. A prime must have exactly two different factors, and 1 has only one.

How do you find the prime factorisation of a number?

Keep dividing by the smallest prime that divides it (2, then 3, then 5, and so on) until you reach 1, or draw a factor tree.

How can I quickly check if a number is prime?

Test the primes up to its square root. For a number below 100, testing 2, 3, 5 and 7 is enough.

Prime numbers and factorisation

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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