- #1

jack476

- 328

- 125

## Homework Statement

The problem is exactly as stated in the title: "Prove that 2

^{55}+ 1 is divisible by 33". This problem is an exercise from the Mathematical Oympiad Handbook in the section on factoring sums of powers.[/B]

## Homework Equations

Results found so far in this section:

x

^{n}- 1 = (x-1)(x

^{n-1}+ x

^{n-2}+...+x

^{2}+x + 1)

x

^{3}+ 1 = (x+1)(x

^{2}- x + 1)

x

^{3}+ y

^{3}= (x+y)(x

^{2}- xy +y

^{2})

## The Attempt at a Solution

So far, I have that 33 = 32 + 1 = 2

^{5}+ 1 = (2+1)(2

^{4}- 2

^{3}+ 2

^{2}-2 + 1) and that 2

^{55}+ 1 = (2+1)(2

^{54}-2

^{53}+2

^{52}+...-2

^{3}+ 2

^{2}-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).