CAT 2024 Slot 2 · Quant · Question 3

Write m in terms of its own prime factors, say m={2}^{a}\times{3}^{b}.

Then {m}^{n}={2}^{an}\times{3}^{bn}, and comparing with the given value, an=25 and bn=40.

So n divides both 25 and 40, which means n divides \gcd \left( 25,40 \right)=5.

The condition n>1 then forces n=5.

That gives a=\frac{25}{5}=5 and b=\frac{40}{5}=8, so m={2}^{5}\times{3}^{8}.

Computing, {2}^{5}=32 and {3}^{8}=6561, so m=32\times 6561=209952.

Finally m-n=209952-5=209947, which is the third option.

Shortcut. The exponent n must divide every exponent on the right, so n∣\gcd \left( 25,40 \right). Once n=5 is forced, m is just the fifth root, read off by dividing exponents.