Question

Find the smallest number to be multiplied with 137592, so that the product is a perfect cube. And also, find the cube root of the resulting product.

• A. 2000, 435
• B. 1183, 546
• C. 1200, 564
• D. 2833, 645

Answer 1

The correct option is B 1183, 546

Given : Number is 137592
Prime factorisation of 137592 :
$\begin{array}{cc}2& 137592\\ 2& 68796\\ 2& 34398\\ 3& 17199\\ 3& 5733\\ 3& 1911\\ 7& 637\\ 7& 91\\ 13& 13\\ & 1\end{array}$
$137592=2\times 2\times 2\times 3\times 3\times 3\times 7\times 7\times 13$

The number 7 and 13 should be multiplied once and twice respectively to make 137592 a perfect cube.

$\therefore$ The smallest number by which 137592 must be multiplied
$=7\times 13\times 13=1183$
Hence, $137592\times 1183$ is a perfect cube.

Now,
$137592\times 1183=2\times 2\times 2\times 3\times 3\times 3\times 7\times 7\times 13\times 7\times 13\times 13$
$=(2^3\times 3^3\times 7^3\times {13}^3)$
$=(2\times 3\times 7\times 13{)}^3$

The cuberoot of $137592\times 1183=\sqrt[3]{3137592\times 1183}=\sqrt[3]{32^3\times 3^3\times 7^3\times {13}^3}=2\times 3\times 7\times 13=546$