how to calculate the modulo of a negative number
It's calculated exactly like the mod of a positive number. In arithmetic modulo cc, we seek to express any xx as qc+rqc+r, where rr must be a non-negative integer.
Why don't we test it out with an example?
Take −100−100 mod 8=48=4. This is because of 8⋅−13=−1048⋅−13=−104. The remainder is 44.
So now let's take (37−54)(37−54) mod 55. It's equal to −17−17 mod 5=35=3. Substitute in and do the computation: Method 11 gives 33, which is what we want, and method 22 gives −2−2, so the correct approach is method 11