1. ## find remainder

Find r when 319^566 is divided by 37.

I can get it down to (11^2)^283*(29^2)^283.

2. Hello, frankdent1!

Are you allowed to use Modulo Arithmetic?

Find the remainder when $\displaystyle 319^{566}$ is divided by $\displaystyle 37.$

We find that: .$\displaystyle 319^6 \:\equiv \:-1 \pmod{37}$

Then: .$\displaystyle 319^{566} \:=\:319^{6\cdot94 + 2} \:=\: (319^6)^{94}\cdot319^2$

Hence: .$\displaystyle (319^6)^{94}\cdot319^2 \:\equiv \-1)^{94}\cdot319^2 \pmod{37}$

. . . . . . . . . . . . . . .$\displaystyle \equiv\:319^2 \pmod{37}$

. . . . . . . . . . . . . . .$\displaystyle \equiv\:101,761 \pmod{37}$

. . . . . . . . . . . . . . .$\displaystyle \equiv \:11 \pmod{37}$