#1**+1 **

Well, let's notice something interesting about this problem.

The number two has only two residues. 0 and 1.

If the number is odd, then the residue is 1. If the number is even, then the residue is 0.

We just have to figure out whether or not \(n \equiv 43 \pmod{165}\)is even or odd.

We can easily figure this out though.

Let's note that since n is 43 mod 165, we can write n as

\(165x+43 =n\) where x is a random integer.

In this case, when x is even, then n is odd.

If x is odd, then n is even.

so in turn, both residues are completely possible for the final result.

Thus, our answer is \(0,1\)

Thanks! :)

NotThatSmart Jul 23, 2024