remainder when common term exists in numerator and denominator and congruence relation

Question

remainder of $\dfrac{10}{3}$ $=1$

remainder of $\dfrac{2×10}{2×3}$ $=2×1$

remainder of $\dfrac{3×10}{3×3}$ $=3×1$

Therefore, can we generalize this as

remainder of $\dfrac{a×b}{a×c}$ $=a$ × remainder of $\dfrac{b}{c}$

I know only the basics of Congruence relation. If the above property is true, how can I write it using Congruence Relation? To my understanding, this can be written as the following . Please correct me if I am wrong.

$\dfrac{3×10}{3×3} \equiv 3×\dfrac{10}{3} \pmod 3$

Answer 1

What you have stated can be written as

If $b \equiv N \mod c \iff a*b \equiv aN \mod a*c$.