Just want some confirmation on what i've done -number theory's my weakest area by far.

1. Calculate 7^23 (mod 10)

MY ANSWER: 7=17 (mod 10)

7^2 = 289 = 9(mod 10)

so 7^22 = (7^2)^11 = 9 (mod 10)

so 7^23 = 7 x 7^22 = 17 x 9^11 = 9(mod 10)

(note: these are meant to be equivalent rather than equals signs)

2. Solve 8x = 12 (mod 20)

MY ANSWER:

gcd(8,20) = 4

4 divides 12 so there are 3 solutions.

A solution is x = 14 (by observation).

Solutions differ by 20/4 = 5.

So 9 and 4 are other solutions as general solutuons given by x= u - (n/d)t