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In Number Theory, I have seen many a times the following property being used though I don't know what it's called formally (that makes it harder to Google):

(a * b) (mod n) = (a mod n) * (b mod n)

What is this called and how can it be formally proved?

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    It's the *definition* of multiplication modulo $n$, after it has been shown it does not depend on the choice of the representatives in each class. – 2017-02-03

1 Answers 1

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If $a=xn+a'$ and $b=yn+b'$, where all are whole numbers, then $a'=a\text{ mod }n$ and $b'=b\text{ mod }n$. It's then easy enough to multiply them:

$$ab=(xn+a')(yn+b')=(xyn+xb'+ya')n+a'b'$$

Take mod $n$ of both sides, and it should then be clear that

$$ab\equiv a'b'\pmod n$$