The remainder when 3²⁰²² is divided by 5 is

Q: Wait, why do we reject the negative remainder $-1$?

In modular arithmetic, the remainder $r$ must satisfy the condition $0 \le r < |b|$, where $b$ is the divisor. Since our divisor is $5$, the remainder must be in the set $\{0, 1, 2, 3, 4\}$. A result of $-1$ is mathematically equivalent to $4$ because $-1 + 5 = 4$. We simply add the divisor to the negative result to bring it into the standard positive range.

Q: Why can't we just use the cyclicity of powers of $3$?

You absolutely can! The powers of $3$ modulo $5$ are $3^1 \equiv 3$, $3^2 \equiv 4$, $3^3 \equiv 2$, $3^4 \equiv 1$, and then the cycle repeats every $4$ powers. Since $2022 = 4 \times 505 + 2$, the remainder is the same as $3^2 \equiv 4$. However, the Binomial Theorem approach is a more powerful, general technique that works even when the cycle length is not easily identifiable.

Q: Is it always possible to use the Binomial Theorem for these types of problems?

Yes, it is a very robust strategy. Whenever you have a base $a$ and a divisor $b$, if you can find a small integer $k$ such that $a^n \approx m \cdot b + k$, you can rewrite the base as $(m \cdot b + k)$ and use the Binomial Theorem to expand it. All terms containing $b$ will vanish modulo $b$, leaving you with only the constant term $k^n$ to evaluate.

Question

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

Options:

(A) 1

(B) 2

(C) 3

(D) 4

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a colossal number: $3^{2022}$. Your task is to find the remainder when this titan is divided by $5$.

If you try to calculate this directly, you will be lost in a sea of digits. In the realm of mathematics, we do not fight brute force with brute force; we fight it with elegance.

### The Golden Link

The first step in our journey is to observe the base, $3$, and the divisor, $5$. Let us look at the powers of $3$:
$3^1 = 3$
$3^2 = 9$

Look at $9$. It is tantalizingly close to $10$, and $10$ is a multiple of $5$. This is our golden link.

We can rewrite our massive expression $3^{2022}$ by using the exponent rule $a^{mn} = (a^m)^n$. Since $2022 = 2 	imes 1011$, we can write:

$$3^{2022} = (3^2)^{1011} = 9^{1011}$$

Now, instead of dealing with $3^{2022}$, we are dealing with $9^{1011}$.

### The Binomial Bridge

We use the Binomial Theorem as our magnifying glass. We rewrite $9$ as $(10 - 1)$. Our expression becomes $(10 - 1)^{1011}$.

When we expand this using the binomial expansion, we get:

$$(10 - 1)^{1011} = \binom{1011}{0}10^{1011} - \binom{1011}{1}10^{1010} + \dots + (-1)^{1011}\binom{1011}{1011}1^{1011}$$

Every single term in this expansion, except for the very last one, contains a power of $10$. Since $10$ is a multiple of $5$, every one of those terms is also a multiple of $5$.

We can group all these terms together and call them $5m$, where $m$ is some integer. Our expression has now collapsed into $5m + (-1)^{1011}$.

### The Final Adjustment

We are almost at the finish line. We have $(-1)^{1011}$. Since $1011$ is an odd number, a negative base raised to an odd power remains negative.

Thus, $(-1)^{1011} = -1$. Our expression is now $5m - 1$.

In the world of remainders, we must have a positive value between $0$ and $4$. A remainder of $-1$ is equivalent to being one step behind the finish line.

To get to the positive remainder, we simply add the divisor, $5$:

$$5m - 1 = 5m - 5 + 4 = 5(m - 1) + 4$$

This is in the form $5k + 4$, where $k = m - 1$. Therefore, the remainder is **4**.

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

Select Answer:

Visualized Solution

The Problem Statement

Strategy for Large Exponents

Finding the Closest Multiple

Rewriting the Expression

Base Transformation

Introducing Binomial Expansion

Expanding $(10 - 1)^{1011}$

Grouping Multiples of $5$

Evaluating the Last Term

Adjusting for a Positive Remainder

The Final Remainder

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing the Setup

The Golden Link

The Binomial Bridge

The Final Adjustment

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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; } The remainder when 32022 is divided by 5 is