question about logically equivalence

The proposition is "(a2 + 3) mod 4 = 0" => "a is an odd number"

Assume a is an even number, i.e. a = 2N for an integer N,
a2 + 3
= (2N)2 + 3
= 4N2 + 3
which is an odd number

so we have "a is an even number" => "a2 + 3 is an odd number"
i.e. "(a2 + 3) is an even number" => "a is an odd number"
in particular, "(a2 + 3) is divisible by 4" => "a is an odd number"