Question

The smallest number by which 3600 can be divided to make it a perfect cube is:
A. 9
B. 50
C. 300
D. 450

Answer 1

Hint: A perfect cube is an integer which is equal to another integer raised to the power 3. For example 125 is a perfect cube as it can be written as $ {5^3} $ , 5 raised to the power 3. So divide 3600 by every given option and if the result obtained can be written as a perfect cube then that particular option is our answer.

Complete step-by-step answer:
We are given to determine the smallest number by which 3600 can be divided to make it a perfect cube.
First one is 9, when 3600 is divided by 9 we get $ \dfrac{{3600}}{9} = 400 $ . But 400 is a perfect square of 20 but not a perfect cube.
Second one is 50, when 3600 is divided by 50 we get $ \dfrac{{3600}}{{50}} = 72 $ . 72 cannot be written as a perfect cube.
Third one is 300, when 3600 is divided by 300 we get $ \dfrac{{3600}}{{300}} = 12 $ . 12 is not a perfect cube.
Fourth one is 450, when 3600 is divided by 450 we get $ \dfrac{{3600}}{{450}} = 8 $ . 8 can also be written as 2 raised to the power 3.
This means that 8 is a perfect cube of 2. So 3600 must be divided by 450 to make it a perfect cube
So, the correct answer is “Option D”.

Note: Another approach
First we have to factorize 3600.
3600 can also be written $ 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 $
As we can see in the prime factorization of 3600, there is a repeated factor 2 for four times and remaining factors are repeated less than 3 times.
For a perfect cube there must be a repeating factor for 3 times.
So 3600 must be divided by $ 2 \times 3 \times 3 \times 5 \times 5 $ to give a perfect cube $ 2 \times 2 \times 2 = 8 $
And $ 2 \times 3 \times 3 \times 5 \times 5 = 450 $
So 3600 must be divided by 450 to give a perfect cube 8.