If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)²⁰²² is divided by 8

Q: Why did we rewrite $3$ as $4-1$?

Because $3$ is $4-1$. This allows us to use the binomial expansion $(4A-1)^n$, where the terms are either multiples of $4$ or powers of $-1$. This makes the modulo $8$ analysis trivial because we can easily see which terms are multiples of $8$.

Q: Why do we ignore terms with $(4A)^2$ and higher?

Because $(4A)^2 = 16A^2$. Since $16$ is a multiple of $8$, $16A^2$ is divisible by $8$. Any higher power of $4A$ will also be a multiple of $8$ (e.g., $(4A)^3 = 64A^3$, which is $8 \times 8A^3$). Thus, all these terms vanish when we consider the expression modulo $8$.

Q: What if the remainder was negative?

Remainders are always non-negative by definition. If you get a result like $-1$ during your calculation, you simply add the divisor ($8$) to it to get the positive remainder ($7$). In this specific problem, the final term is $(-1)^{2022}$, which is $+1$, so we don't have to worry about negative remainders.

Question

If the remainder when $x$ is divided by 4 is 3, then the remainder when $(2020 + x)^{2022}$ is divided by 8 is

Sigma Percentile · Accepted Answer

### The Beauty of Number Theory

My dear student, welcome to a journey through the elegant world of number theory. Today, we are going to unravel a problem that might look like a mountain of algebra at first glance, but is actually a beautifully structured puzzle.

We are asked to find the remainder when $(2020 + x)^{2022}$ is divided by $8$, given that $x$ leaves a remainder of $3$ when divided by $4$. Let's peel back the layers together.

### Decoding the Clue

First, let's translate the condition $x \equiv 3 \pmod 4$ into the language of algebra. By the division algorithm, we can write $x = 4k + 3$, where $k$ is an integer.

Now, look at our base: $(2020 + x)$. If we substitute our expression for $x$, we get $2020 + 4k + 3$.

Notice that $2020$ is a multiple of $4$ ($2020 = 4 	imes 505$). So, our base becomes $4(505) + 4k + 3$, which simplifies to $4(505 + k) + 3$.

Let's define $P = 505 + k$. Now, our expression is simply $(4P + 3)^{2022}$.

### The Binomial Transformation

We need to find the remainder when this is divided by $8$. Expanding $(4P + 3)^{2022}$ directly would be a nightmare.

But here is the JEE-style trick: rewrite $3$ as $4 - 1$. Our base becomes $4P + 4 - 1$, which is $4(P + 1) - 1$.

Let $A = P + 1$. Now, the expression is $(4A - 1)^{2022}$. Why is this so much better? Because the Binomial Theorem loves having a $1$ or a $-1$ in the base!

### The Binomial Magic

Let's expand $(4A - 1)^{2022}$ using the Binomial Theorem:

$$ (x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r $$

Our expansion looks like this:

$$ \binom{2022}{0}(4A)^{2022} - \binom{2022}{1}(4A)^{2021} + \dots - \binom{2022}{2021}(4A)^1 + \binom{2022}{2022}(-1)^{2022} $$

Look closely at the terms. Any term containing $(4A)^k$ where $k \ge 2$ is a multiple of $16$ (since $4^2 = 16$). Because $16$ is a multiple of $8$, all these terms are perfectly divisible by $8$ and effectively become zero in our modulo $8$ world.

### The Final Reveal

We are left with only the last two terms:

$$ -\binom{2022}{2021}(4A)^1 + \binom{2022}{2022}(-1)^{2022} $$

The first of these is $-2022 	imes 4A = -8088A$. Since $8088$ is divisible by $8$ ($8088 = 8 	imes 1011$), this term is also a multiple of $8$.

Finally, we have the last term: $\binom{2022}{2022}(-1)^{2022}$. Since $2022$ is an even number, $(-1)^{2022} = 1$.

Thus, the entire expression is some multiple of $8$ plus $1$. The **remainder is 1**. Isn't it satisfying how the complexity just melts away?

Animated Solution for Mathematics - Binomial Theorem: If the remainder when $x$ is divided by 4 is 3, then the remainder when $(2020 + x)^{2022}$ is divided by 8 is

Enter Numerical Value:

Visualized Solution

Understanding the Goal

Decoding the Clue for $x$

Substituting $x$ into the Expression

Simplifying the Base

Factoring out the Common Multiple

Strategic Rewrite for Binomial Expansion

Finalizing the Base Form

Applying the Binomial Theorem

Analyzing Higher Powers for Divisibility

Evaluating the Second-to-Last Term

The Final Remainder

The Sigma Insight: Binomial Expansion for Positive Integral Index

The Beauty of Number Theory

Decoding the Clue

The Binomial Transformation

The Binomial Magic

The Final Reveal

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the remainder when x is divided by 4 is 3, then the remainder when (2020+x)2022 is divided by 8 is