Modular arithmetic of numbers

Question

let us consider two integers a,b that are co prime to a prime number p

Then is there any relation between a%p, b%p and ab%p ? % = modulo operator

Answer 1

In general, if $r_m(a)$ denotes the remainder of $a$ when divided by $m$, you have that $$r_m(ab)=r_m(r_m(a)\cdot r_m(b))$$

Answer 2

$(a \times b)\pmod p = (a \pmod p \times b \pmod p) \pmod p$

The modulo of the product is equal to the product of the modulos modulo $p$.

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