mod question in discrete math

Question

mod problem,

for any positive integer m

m - 1 mod m

Could I say it congruence to m - (m - 1) mod m in next line?

Below is from a real exam, I was rushing to it, so the explanation might not very clear.

Shreyas claims that for all positive integers m > 2, the number m − 1 always has an inverse modulo m. Do you agree? Either prove your answer or give a counterexample

My attempt: I agree,

m - 1 mod m

= m - m + 1 mod m

= 1 mod m, which is multiplicative inverse

gcd(m, m-1) = 1

Answer 1

Here is the proof that m-1 is its own inverse mod m:

(m-1)*(m-1) mod m = (m² - 2m + 1) mod m = 1

I did not understand the question at the beginning of your post.

But let me just say that

m - (m-1) mod m = 1, which is not congruent to (m-1) mod m, except for m = 2.