How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

Question

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?

Answer 1

$$2010^{1020} + 1020^{2010} = (3\cdot 670)^{1020} + (3\cdot 340)^{2010}\\= 3\cdot (3^{1019}\cdot 670^{1020} + 3^{2009}\cdot 340^{2010})$$

So the number is divisible by $3$ and hence the remainder is $0$.

Answer 2

The remainder when divided by 3 is known as the "mod" operation, i.e. $\pmod 3$.

First, note that both $2010$ and $1020$ are divisible by 3:

$2010 \equiv 0 \pmod 3$ $1020 \equiv 0 \pmod 3$

Raise anything that is $0 \pmod 3$ to a positive integral power, and it will remain $0 \pmod 3$ (i.e. it will remain a multiple of 3), and adding two zeroes $\pmod 3$ yields $0 \pmod 3$.

$2010^{1020} + 1020^{2010} \pmod 3 = 0^{2010} + 0^{2010} \pmod 3 = 0 + 0 \pmod 3 = 0 \pmod 3$