Question

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

Answer 1

$\underline{\text{Given:}}$

$\text{198n is a perfect cube}$

$\underline{\text{To Find:}}$

$\text{The smallest possible integer for n.}$

$\underline{\text{Explanation:}}$

$\text{For a number to be a perfect cube, }$

$\text{all its prime factors must be a perfect cube.}$

$\boxed {\textbf{Solution}}$

$\underline{\text{Find the prime factors for 198:}}$

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

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

$\underline{\text{Find n:}}$

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

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

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

$n = 1452$

$\boxed {\boxed{\textbf{Answer: n = 1450}}}$