1. The problem statement, all variables and given/known data The problem is exactly as stated in the title: "Prove that 255 + 1 is divisible by 33". This problem is an exercise from the Mathematical Oympiad Handbook in the section on factoring sums of powers. 2. Relevant equations Results found so far in this section: xn - 1 = (x-1)(xn-1+ xn-2 +...+x2 +x + 1) x3 + 1 = (x+1)(x2- x + 1) x3 + y3 = (x+y)(x2- xy +y2) 3. The attempt at a solution So far, I have that 33 = 32 + 1 = 25 + 1 = (2+1)(24 - 23 + 22 -2 + 1) and that 255 + 1 = (2+1)(254-253+252+...-23 + 22 -2 + 1) I do not know how to proceed from here. Presumably I'd want to show that the quotient of the two polynomials is an integer, but I don't know how to do that besides polynomial long division, and I doubt that's what the problem is trying to emphasize (and with so many terms, it would probably take forever, if it's even possible for polynomials of orders 5 and 55- not that I'd want to anyway, with so many terms).