Question

Sum of the digits of the smallest number by which $1440$ should be multiplied so that it becomes a perfect cube is _____.

• A. $4$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is B $6$

Prime factorising $1440$, we get,

$1440=2\times 2\times 2\times 2\times 2\times 3\times 3\times 5$

$=2^5\times 3^2\times 5^1$.

We know, a perfect cube has multiples of $3$ as powers of prime factors.

Here, number of $2$'s is $5$, number of $3$'s is $2$ and number of $5$'s is $1$.

So we need to multiply another $2$, $3$ and $5^2$ in the factorization to make $1440$ a perfect cube.

Hence, the smallest number by which $1440$ must be multiplied to obtain a perfect cube is $2\times 3\times 5^2=150$.

$\therefore$ The sum of digits of the smallest number is $=1+5+0=6$.

Hence, option $B$ is correct.