Discrete maths congruences question

Question

"By the definitions of modm and of congruence modulo m, we know that a ≡ (a mod m) (mod m) and b ≡ (b mod m) (mod m)."

why is a ≡ (a mod m) (mod m), is there a proof for this?

Answer 1

By definition, $a\mod m$ is the largest integer $k$ less than $m$ such that $m\mid(a-k)$--that is, $a\equiv k\pmod{m}$. So, by definition, $a\equiv(a\mod m)\pmod{m}$.