- #1
tomwilliam
- 145
- 2
Homework Statement
Show the remainder when 43^43 is divided by 17.
Homework Equations
$$43 = 16 \times 2 + 11$$
$$a^{p-1}\equiv1\ (mod\ p)$$
The Attempt at a Solution
I believe I can state at the outset that as:
$$43\equiv9\ (mod\ 17)$$
Then
$$43^{43}\equiv9^{43}\ (mod\ 17)$$
and that I can rewrite this as:
$$9^{43}=\left(9^{2}\right)^{16}\times9^{11}$$
Then applying Fermat's Little Theorem we get:
$$\left(9^{2}\right)^{16}\equiv1\ (mod\ 17)$$
So that
$$43^{43}\equiv 1\times9^{11}\ (mod\ 17)$$
But I'm unsure where to go from here.
Any help appreciated