If $5400mn = k^4$ , where $m$ , $n$ , and $k$ are positive integers, what is the least possible value of $m + n$ ?

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Comments

AmeliaJul 6

Took me a sec to see that I need to prime factorize 5400 first. Once I got 2^3 * 3^3 * 5^2, the rest clicked. Clever question.

ShantanuJul 9

Same here. The exponents not being multiples of 4 is the whole trick.

YashikaJul 9

Yeah, I stared at it way too long before factoring haha.

DiegoJul 14

Wait, do we need m and n to be as small as possible individually, or just their sum? Because the exponents might force us to add more than we think.

RobertoJul 22

The question only cares about m+n, so you can distribute the missing factors however you want.

AryamanJul 27

Right, but you still want the minimum total, so don't overpad any single prime.

TanveerApr 29

Got 20 but I honestly guessed between C and D. Is there a quick way to check without fully listing the factors?

AshishMay 8

Just check each prime's exponent mod 4 and add what's missing. That's the fast way.

JavierMay 15

605-655 feels right for this one. Not super hard but definitely easy to mess up if you forget to account for all three primes.

MarceloMay 25

Agreed, I forgot the 5 at first and got the wrong sum.

What this question tests

Your grasp of exponent rules — multiplying and dividing powers, power-of-a-power, and negative/zero exponents.

Adding exponents when multiplying bases that have the same exponent, or confusing (x^a)^b with x^(a·b).

If 5400mn=k45400mn = k^45400mn=k4, where mmm, nnn, and kkk are positive integers, what is the least possible value of m+nm + nm+n?