Question

By what smallest number should 3600 be multiplied so that the quotient is a perfect cube. Also find the cube root of the quotient.

Answer 1

Prime factors of 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

Grouping the factors into triplets of equal factors, we get

3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

2	3600
2	1800
2	900
2	450
3	225
3	75
5	25
5	5
	1

We know that, if a number is to be a perfect cube, then each of its prime factors must occur thrice.

We find that 2 occurs once 3 and 5 occurs twice only.

Hence, the smallest number, by which the given number must be multiplied in order that the product is a perfect cybe = 2 × 2 × 3 × 5 = 60

Also, product = 3600 × 60 = 216000

Now, arranging into triplets of equal prime factors, we have

216000 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5

Taking one factor from each triplets, we get

${\displaystyle \sqrt[3]{216000}}$ = 2 × 2 × 3 × 5 = 60