Given a==1(mod 7), b== 2 (mod7), and c == 6 (mod7), what is the remainder when a^81 b^91 c^27 is divided by 7?

means there is an integer such that , or .

Raising both sides to an arbitrary integer power, we have



Notice that each term in the expansion on the right is a multiple of 7 when , which means modulo 7, the right side reduces to 1. Therefore if , then as well.

More generally, the remainder of a number upon dividing by 7 will be determined by the constant term (independent of ) in the binomial expansion, because any term with a contributing factor of necessarily is a multiple of 7.

You then have





Now,



Recall that for and , we have , which means we can determine the remainder above by multiplying the remainders given by and .

In particular, if , then



Now, we get by this property in conjunction with Fermat's little theorem that



















So we obtain