# Calculating potencies with modulo without a calculator

I want to calculate

$$119^{7} \text{mod}~ 143$$

I got this one step from a solution sheet, which I don't understand: $$119^{7} ~\text{mod}~ 143 = (-12 * 2)^{7} ~\text{mod}~143$$ Still getting this, because $-24$ is the same like 119. But now the strange thing: $$(-12 * 2)^{7} ~\text{mod}~143 = -12 * 128 ~\text{mod}~143$$ How does this work? I mean I understand, that $2^{7}$ is $128$, but why is there just $-12$ instead of $-12^{7}$?

It ends with this (which I understand): $$= 12 * 15 ~\text{mod}~143 = 180 ~\text{mod}~ 143 = 37$$

Can someone explain me this one 'strange' step for me? Thank you so much.

Note that $$(-12)^7 \equiv ((-12)^2)^3(-12) \equiv 1^3(-12) \equiv -12 \pmod {143}$$ since $12^2 = 144$.

$$-12\equiv-1\pmod{11}$$ so $$(-12)^7\equiv-1\equiv-12\pmod{13}.$$ $$-12\equiv1\pmod{13}$$ so $$(-12)^7\equiv1\equiv-12\pmod{13}.$$ By the Chinese Remainder Theorem, $$(-12)^7\equiv-12\pmod{143}.$$

Somehow the text is assuming it must be obvious that $(-12)^7 \equiv -12 \mod 143$. So we should ask ourselves why that would be.

$143 = 11*13 = (12-1)*(12 + 1) = 12^2 -1 = 144$. Therefore $(\pm 12)^2 \equiv 1 \mod 143$ and $(-12)^7 = -12(-12)^6 \equiv -12 \mod 143$.