The following question showed up in an aptitude test I took.
If $N = 2197^P × 144^2 × 2^R × 3^S$ is the perfect cube of a natural number, where $P$, $R$ and $S$ are distinct positive integers, then find the minimum value of $(P + R + S)$.
I solved it as follows: My solution
However, the correct answer is 6 as per the answer key released. Could anyone please tell where am I going wrong?
Here is the solution provided by the testing agency: Provided solution
$$2197 = 13^3$$ $$144 = 2^4 3^2$$
exponent of $13$ is $3P$
exponent of $2$ is $4 + R$
exponent of $3$ is $2+S$
So $P$ is $1,2,3,4,5...$
$R$ is $2,5,8,...$
$S$ is $1,4,7,...$
If $P$ is $1,2$ we need a second choice, best would be $S=4,$ but then $P+R \geq 3$ regardless, so sum is at least $7$
With $P=3, R=2, S=1 $ we get sum down to $6,$ and we cannot do better since $R+S \geq 3$