The remainder when $$7^{2022} + 3^{2022}$$ is divided by 5 is

Solution :

We need to find remainder when

$$7^{2022}+3^{2022}$$

is divided by $$5$$.

Now,

$$7 \equiv 2 \pmod 5$$

Therefore,

$$7^{2022} \equiv 2^{2022} \pmod 5$$

Also,

Powers of $$2$$ modulo $$5$$ repeat every 4 :

$$2^1 \equiv 2$$

$$2^2 \equiv 4$$

$$2^3 \equiv 3$$

$$2^4 \equiv 1 \pmod 5$$

Since,

$$2022 \equiv 2 \pmod 4$$

therefore,

$$2^{2022} \equiv 2^2 \equiv 4 \pmod 5$$

Now for $$3^{2022}$$ :

Powers of $$3$$ modulo $$5$$ also repeat every 4 :

$$3^1 \equiv 3$$

$$3^2 \equiv 4$$

$$3^3 \equiv 2$$

$$3^4 \equiv 1 \pmod 5$$

Again,

$$2022 \equiv 2 \pmod 4$$

Hence,

$$3^{2022} \equiv 3^2 \equiv 4 \pmod 5$$

Therefore,

$$7^{2022}+3^{2022} \equiv 4+4$$

$$\equiv 8$$

$$\equiv 3 \pmod 5$$

Final Answer :

$$3$$