Prove that 12 is not a divisor of:

Question

Prove that $12$ is not a divisor of $ 10 ^ 7-10 $.

I did like this:

I rewrote it as $ 9. \frac {10 ^ 6-1} {9} .10 $ and analyzed that this product has no factor $ 2 ^ 2$. Are there any more elegant methods?

Answer 1

One way:

Say $n>1$, then $$10^n-10 = 10\underbrace{(10^{n-1}-1)}_{2k+1} = 2\cdot 5\cdot (2k+1)$$ so it is not divisible by $4$.

Answer 2

You could write

$$10^7 - 10 = (12 - 2)^7 - (12 - 2)$$

If you expand that out, everything is a multiple of $12$ except $(-2)^7 + 2 = - 126$ which is not a multiple of $12.$

Answer 3

$10^7\equiv 0$, $\;10\equiv 2 \mod 4$, so $\; 10^7-10\equiv -2\equiv 2\mod 4$, in other words, $10^7-10$ is not divisible by $4$, hence not divisible by $12$.