Proving N is a Composite Number: Larson 4.1.8 [SOLVED]

[ehrenfest]

[SOLVED] larson 4.1.8

Homework Statement

Let N be the number which when expressed in decimal notation consists of 91 ones:

1111...1111 = N

Prove that N is a composite number.

Homework Equations

The Attempt at a Solution

If N had an even number n of ones we could use the fact that

$$\sum_{i=0}^n x^i = (1+x)(x+x^3+x^5+...+x^{n-1}) = N$$

evaluated at x=10. I tried doing lots of similar tricks for the odd case but nothing seems to factor completely.

 

[morphism]

Since 91=13*7, then 10^91 - 1 = 0 (mod 10^7 - 1) (and 10^91 - 1 = 0 (mod 10^13 - 1)) - prove this. And since 9N = 10^91 - 1, it follows that N is divisible by (10^7 - 1)/9 (and (10^13 - 1)/9).