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168 = 2³ x 3 x 7

So in order to have a perfect square, all your prime factors must be ___².

Since your question asks to find the smallest value of k, we multiply by the smallest possible values to make the prime factors become __²

You have 3. So just multiply by 3 to obtain 3².
You have 7. So just multiply by 7 to get 7².
You have 2³. So just multiply by 2 to get 2⁴.

Realise that 2⁴
= 2 x 2 x 2 x 2
= 2² x 2²
= (2²)²
= 4²

So x 3 x 7 x 2 → 42.

k = 42

168 x 42 = 2³ x 3 x 7 x 2 x 3 x 7
= 2⁴ x 3² x 7²
= (2² x 3 x 7) x (4 x 3 x 7)
= (4 x 3 x 7)²
= 84²
= 7056


Likewise, √7056 = 84 = 4 x 3 x 7

240 = 2⁴ x 3 x 5

168 = 2³ x 3 x 7

In order for 168n to be a multiple of 240,
168n must contain all the prime factors of 240.

So 168n has to contain 2⁴, 3 and 5.

2⁴ : 168 only has 2³. So multiply by 2
to get 2⁴.

3 : 168 already has one 3.

5 : 168 does not have 5. So multiply by 5.

So x 2 x 5 →x 10

So n = 10

168n
168 x 10 = 1680
= 2³ x 3 x 7 x 2 x 5
= 2⁴ x 3 x 5 x 7
= 240 x 7

Here's a fast way for you to check your working and answer for part b).


168 ÷ 240 = 0.7

168 is 0.7 times of 240. 0.7 is a decimal.

How to make 168 a multiple of 240?

Just multiply by 10.

168 x 10 ÷ 240
= 168 ÷ 240 x 10
= 0.7 x 10
= 7

So n is 10.

You can also see it as shifting the decimal place one time to get 0.7 to become a whole number.