Confused about a Wikipedia explanation on primitive Pythagorean triplets

Question

Link to the Paragraph: "Proof of Euclid's formula" https://en.wikipedia.org/wiki/Pythagorean_triple#:~:text=Proof%20of%20Euclid%27s%20formula

In this paragraph (please read from 4th sentence for context), there's a statement as follows:

As a and b are coprime, at least one of them is odd, so we may suppose that a is odd, by exchanging, if needed, a and b. This implies that b is even and c is odd (if b were odd, c would be even, and c² would be a multiple of 4, while a² + b² would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4)

How can be c² be a multiple of 4 and c be a multiple of 2?

My argument: If both a and b are odd, then there squares are congruent to 1 mod 4 and therefore a² + b² is congruent to 2 mod 4 (which is clearly not a multiple of 4).

Please explain this ambiguity.

Edit 1: Extended argument:

We can agree that if both a and b are odd, then there squares are odd (congruent to 1 (mod 4)) and thus the sum of squares is even (congruent to 2 (mod 4))

I get that they are they to prove that both a and b can't be odd using proof of contradiction, so the two contradictory statements that they provide us are:

1. If both a and b are odd, then a^2 + b^2 is congruent to 2 (mod 4)
2. if b were odd, c would be even (how and why???), and c² would be a multiple of 4**

They are trying to show that one statement concludes that a^2 + b^2 is congruent to 2 (mod 4), whereas c^2 is clearly a multiple of 4, i.e. congruent to 0 (mod 4).

But I want to know the arguments which would imply that c is even (using which they concluded there second statement)