Question

Find the least residue of 7^5 mod 50 without using a calculator.

so far I have 7=7 mod 50 7^5 = 7^5 mod 50 7^2 = 49 7^3=343 since 7^2=(-1)mod50 and 7^3=(-43)mod50 it follows that 7^6=7^2 * 7^3 = (-1)(-43) = 43 feel like this is wrong though could somebody please explain where I have made an error?

Answer 1

7^1 = 7 (mod 50)
7^2 = 49 = -1 (mod 50)
7^3 = 7^2*7 = (-1)*7 = -7 (mod 50)
7^4 = (7^2)*(7^2) = (-1)*(-1) = 1 (mod 50)
7^5 = (7^4)*7 = 1*7 = 7 (mod 50)

So 7^5 = 7 (mod 50)

In other words, if we divided 7^5 over 50, the remainder would be 7. We don't need to worry about the quotient.