show that one of the numbers $1,11,\cdots, 11\cdots 1$ is divisible by p [duplicate]
Solution 1:
For $p = 3$ this is true.
Assume $p \neq 2,3,5$. Note that $$\underbrace{1...1}_{p-1\text{ times}} = \frac{10^{p-1}-1}{9} \equiv 0 \mod p.$$
For $p = 3$ this is true.
Assume $p \neq 2,3,5$. Note that $$\underbrace{1...1}_{p-1\text{ times}} = \frac{10^{p-1}-1}{9} \equiv 0 \mod p.$$