Animated Solution for Mathematics - Binomial Theorem: The remainder when $7^{2022}+3^{2022}$ is divided by 5 is:

Imagine you are sitting in the examination hall, and you see the expression $7^{2022}+3^{2022}$ divided by $5$. Your first instinct might be panic, but you are not meant to calculate the gargantuan number itself.

In the realm of JEE Advanced, we look for the essence and behavior of the expression. We will strip away the complexity and find the remainder using the elegant language of modulo arithmetic.

When we divide by $5$, we are essentially asking: "Where does this number land on a clock with $5$ hours?" We do not care about how many full rotations the number makes; we only care about the final position.

This is why we use the notation $a\equiv b(\mathrm{mod}n)$. It tells us that $a$ and $b$ are essentially the same when viewed through the lens of the divisor $n$. Our goal is to simplify the bases $7$ and $3$ until they are easy to handle.

We want to express our bases as $5k\pm 1$. This is because $1$ and $-1$ are the most powerful numbers in exponentiation; $1$ raised to any power is $1$, and $-1$ raised to an even power is $1$, while $-1$ raised to an odd power is $-1$.

For the base $7$, we observe:

Similarly, for the base $3$, we observe:

We have found our key to neutralizing the massive exponent $2022$.

Now we rewrite our original expression by utilizing the properties of exponents. We can express $7^{2022}$ as:

Since $1011$ is an odd number, we find:

We apply the exact same logic to $3^{2022}$:

We are left with the sum of our two results:

In the world of remainders, we prefer positive integers. If you are $2$ steps behind a multiple of $5$, you are exactly $3$ steps ahead of the previous multiple.

Mathematically, we add the divisor to normalize the result:

The remainder is $3$. You have just tamed a monster of an expression using nothing but logic and the power of $-1$.