Solution of the Week #291 - Divisible by 32

n=2

Since 26 and 6 are both even, if n is greater than or equal to 5, both parts will independently be divisible by 32, so be only need to consider n = 1,2,3,4.

When n=1, it is obvious that 26+6=32.

When n=3, we note that (x^3 + y^3) can always be factorised into

(x+y)(x^2–xy+y^2), so letting x=26 and y=6 shows that this is divisible by 32.

When n=4, each part will be an odd number multiplied by 2^4, so will naturally have a remainder of 16 when divided by 32. Putting both of them together gives a multiple of 32.

That just leaves n=2. Unlike all of the other numbers, there is no inherent reason why this would be divisible by 32, and simply checking shows that it isn’t.