Which number is divisible by $3^6$?

Question

Which number is divisible by $3^6$?

$30^2\times 75^4$

$15^2\times162$

$30 \times 18^2$

$6^2\times 30^3$

I cannot show any attempts that I have tried because I don't even know where to start. I know there must be a simpler way to find out.

Answer 1

To be divisible by $3^{6}$ it must have six factors of 3 in it.

$30^{2}\times 75^{4} = (3\times 10)^{2}\times (3\times 25)^{4} = 3^{2}\times 10^{2}\times 3^{4}\times 25^{4} = 3^{6}\times 10^{2}\times 25^{4}$ so this has six factors of 3 and is divisible by $3^{6}$.

Note that I could simplify this number and any other number into a product of all of its prime factors (and their powers) and this will immediately tell me pretty much everything about its divisbility.

Answer 2

Hint: Find the prime factors of the numbers in each of the choices and regroup terms. It will then be easy to see which is divisible by $3^6$.