Prime factorization and modulo for large exponents

Question

Given that $7xy7xy...7xy$ is a multiple of $143$ where there are $2008$ occurrences of $7xy$. Find the values of $x$ and $y$.

I know that $143$ is a semiprime with factors $1, 11, 13, 143$. How should I proceed to break down the large exponent in order to check if the modulo would be zero?

Answer 1

Hint: $1001= 7 \times 143$.

Your number is $(700+10a+b)\times 1001 \times (\sum_{i=0}^{1003} 10^{6i})$.

So any values of $a$ and $b$ will do.

Answer 2

Use congruences: $\;10^3\equiv -1\mod 143$, so $$\underbrace{7ab\,7ab\,\dots\,7ab}_{2008\text{ times}}=\sum_{k=0}^{2007}7ab\,(10^3)^k\equiv7ab\sum_{k=0}^{2007}(-1)^k=0,$$ because there is an even number of $(-1)^k$ terms, so any values for $a$ and $b$ are fine.