Question

What is the smallest number which should be multiplied to $231525$ to make it a perfect cube.
A) 5
B) 3
C) 7
D) 21

Answer 1

Hint: To solve this question we need to have the knowledge of Prime Factorisation. The first step to solve the problem is to find all the prime factors of the number given. The number will be written as the product of prime factors. Then we will check the number which should be multiplied so that the number given turns into a perfect cube.

Complete step by step answer:
The question ask us to find the value which should be multiplied to the number $231525$ so that the number becomes a perfect cube. The first step will be to find the prime factors of the number with the help of prime factorization of the number, $231525$. After prime factorization we see that the number $231525$ has $3,5$ and $7$ as its prime factors.
On writing the number $231525$ in terms of its prime factor we get:
$\Rightarrow 231525=3\times 3\times 3\times 5\times 5\times 7\times 7\times 7$
$\Rightarrow 231525={{3}^{3}}\times {{5}^{2}}\times {{7}^{3}}$
Through the above expression we see that to make a number perfect cube we are supposed to multiply the number with $5$ as in the product we see that if $5$ is multiplied to the number then it will occur thrice in the product making the number perfect cube. So on multiplying the number with $5$we get:
$\Rightarrow 3\times 3\times 3\times 5\times 5\times 5\times 7\times 7\times 7=231525\times 5$
$\Rightarrow {{3}^{3}}\times {{5}^{3}}\times {{7}^{3}}=1157625$
$\Rightarrow {{105}^{3}}=231525\times 5$
$\therefore $ The number $231525$ is multiplied with $5$ to make the number a perfect cube number.

So, the correct answer is “Option A”.

Note: While finding the prime factors of a number we are supposed to write the factors carefully. The number should be written as the product of its factors. A number is said to be cube of a number when it is multiplied thrice to itself. Mathematically it is written as ${{a}^{3}}=a\times a\times a$.