Question

Find the smallest possible integer,n, such that 198n is a perfect cube.

Answer 1

Given:

perfect cube of 198n

To find:

Find the smallest possible integer.

Solution:

Calculating the prime factor of 198:

$\to 198= 2\times 3\times 3 \times 11$

          $= 2 \times 3^2\times 11$

$\to 198= 2\times 3^2 \times 11......(i)$

In the given equation we have 198n. So prime factor of 198n:

$\to 198n=2^n \times 3^n\times 11^n$

if n= 3:

$\to 198n=2^3 \times 3^3\times 11^3.......(ii)$

divide the equation ii by i:

$\to n= 2^2\times 3\times 11^2$

       $= 4\times 3\times 121\\\\=12\times 121\\\\=1452$

The final value of n is: 1452