Problem 5

Let

a,b,c

be integers satisfying

0<a<c-1

and

1<b<c

. For each

k

0\le k\le a

, let

r_k

0\le r_k<c

, be the remainder of

kb

when divided by

c

. Prove that the two sets

\{r_0,r_1,r_2,\ldots,r_a\}

and

\{0,1,2,\ldots,a\}

are different.

Step 3 of 5: Factor the polynomial

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(x^b-1)(x-1)f(x)=F(x)=x^{ab+b+1}+x^b+x^{a+1}-x^{ab+b}-x^{a+b+1}-x

Detailed analysis

Using the two finite geometric sums gives (x^b−1)(x−1)f(x)=F(x), where F has the six displayed monomials. Since x^c−1 divides f(x), it also divides F(x).