Use prime factors to explain why $135\times200$ is a perfect cube. The lowest common multiple of $x$ and $135$ is $2\times3^{4}\times5\times7$. Find the smallest possible value of $x$. Answer $x$ = ...

1. Write 135 and 200 as products of prime factors.

$135=3^3\times5$

$200=2^3\times5^2$

2. Combine the prime factors and explain why the product is a perfect cube.

$135\times200=(3^3\times5)(2^3\times5^2)$

$=2^3\times3^3\times5^3$

Since all the powers are multiples of 3,

$2^3\times3^3\times5^3=(2\times3\times5)^3=30^3$.

Therefore $135\times200$ is a perfect cube.

1. Compare the prime factors of $135$ with the given LCM and choose the smallest possible $x$.

$135=3^3\times5$

Given
${\rm LCM}(x,135)=2\times3^4\times5\times7.$

The factor $5$ is already supplied by $135$, but $x$ must supply $2$, $3^4$ and $7$ so that the LCM has these powers.

Smallest possible
$x=2\times3^4\times7=2\times81\times7=1134.$