GRE Quant: How to Find Remainder with Exponents

Remainder questions with exponents focus on finding the remainder of a power expression when divided by a given number. You work with the base and the exponent together, track how the value behaves under division, and use that structure to determine the required remainder with precision. Very interesting and layered questions get developed and tested around the concept of finding remainders with exponents,

The following conceptual video, part of our comprehensive online GRE course, covers how to find remainders with exponents for GRE Quant. It explains the core idea, outlines how this concept commonly appears on the GRE Quantitative section, and then applies it to multiple GRE-style problems so you gain first-hand experience using it. Take your time to absorb the concepts and apply them in your further GRE practice on GRE quizzes, GRE sectional mocks, and GRE full mocks.

 

Concept + Worked Example 1

To calculate remainders for large exponents, the most effective strategy is to look for a remainder of 1 or -1 in the base.

Problem Statement

What is the remainder when (8)³⁰⁰ is divided by 7? 

Core Concept and Calculation

The goal is to simplify the base relative to the divisor before dealing with the exponent.

• First, divide the base (8) by the divisor (7) to find the individual remainder.
• 8 divided by 7 leaves a remainder of 1 (expressed as 8 % 7 = 1).
• Because the base leaves a remainder of 1, the entire expression 8^300 % 7 can be simplified by replacing the base with its remainder.
• This transforms the problem into (1)^300 % 7.
• Since 1 raised to any power is 1, the expression becomes 1 % 7. 

Final Result

The remainder is 1.

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Worked Example 2

Remainder Theorem for Exponents

When calculating the remainder of a large exponential expression, the goal is to find a base that results in a remainder of 1 or -1 when divided by the divisor. This simplifies the calculation because any power of 1 is 1, and any power of -1 is either 1 (for even exponents) or -1 (for odd exponents).

Problem

Find the remainder when (8)³⁰⁰ is divided by 9.

Analyze the base

Divide the base (8) by the divisor (9).

• 8 divided by 9 gives a remainder of -1.

Substitute the remainder

Replace the base with its remainder in the original expression.

• (8)³⁰⁰ mod 9 becomes (-1)^300 mod 9.

Evaluate the power

Since 300 is an even number, (-1)³⁰⁰ equals 1.

• 1 mod 9 = 1.

Final Result

The remainder is 1.

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Worked Example 3

To find the remainder when a number with an exponent is divided by another, you can use the concept of negative remainders to simplify the calculation.

Core Concept: Negative Remainders

When dividing, if the base is one less than the divisor, it can be expressed as having a remainder of -1. If a calculation results in a negative remainder, you must add the divisor to that negative value to find the final, positive remainder.

Problem

Find the remainder when (8)³³ is divided by 9.

Simplify the Base

Divide the base (8) by the divisor (9). Since 8 is 1 less than 9, the remainder is -1. Expression: 8 % 9 = -1 

Apply the Exponent

Replace the base with its remainder and raise it to the original power. Expression: (-1)³³ % 9 

Calculate the Result

Since 33 is an odd number, (-1) raised to the power of 33 remains -1. Expression: -1 % 9 = -1 

Convert to Positive Remainder

Because a remainder cannot be negative in the final answer, add the divisor (9) to the negative result. Calculation: -1 + 9 = 8

Final Answer

The remainder is 8.

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Worked Example 4

Remainders Type III: Exponents Based

To find the remainder of a large exponential expression, the primary goal is to find a power of the base that results in a remainder of 1 or -1 when divided by the divisor. This simplifies the calculation significantly because any power of 1 is 1.

Example Problem

Find the remainder when (2)³⁰⁰ is divided by 7.

Identify a power of the base near the divisor

The base is 2. We know that (2)³ = 8. 

Check the remainder of that power

When 8 is divided by 7, the remainder is 1. 

Rewrite the expression

We can rewrite (2)³⁰⁰ using the power of a power rule: (2)³⁰⁰ = (2³)¹⁰⁰ = (8)¹⁰⁰ 

Apply the remainder rule

8^100 % 7 is equivalent to: (8 * 8 * 8 … 100 times) % 7 

Substitute the remainders

Since 8 % 7 = 1, we replace each 8 with 1: (1 * 1 * 1 … 100 times) % 7 

Final Result

The remainder is (1)¹⁰⁰ = 1

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Worked Example 5

To calculate the remainder of 2 raised to the power of 303 when divided by 9, the following mathematical steps are used:

Rewrite the expression

The term (2)³⁰³ is converted into (2³)¹⁰¹, which equals (8)¹⁰¹.

Expansion

This is expressed as 8 multiplied by itself 101 times, all divided by 9.

Substitution using remainders

Since 8 is equivalent to -1 when considering division by 9 (because 8 – 9 = -1), the expression becomes -1 multiplied by itself 101 times.

Calculation

Because the exponent 101 is an odd number, -1 raised to that power remains -1.

Final conversion

Since a remainder is typically expressed as a positive value, the result -1 is added to the divisor 9 to get the final answer of 8.

The remainder is 8.

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A Conceptual Practice Set…

Correct Answers:

1. 1
2. 16
3. 1
4. 1

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, stepwise written explanation…

 

Question

1. (3)⁵⁰ % 8
2. (2)⁶⁰ % 17
3. (7)¹⁸ % 24
4. (2⁹⁹ × 5⁹⁹)⁹⁹ % 9

 

Explanation

To solve these problems, we use the property of remainders where we find a power of the base that is close to a multiple of the divisor (usually 1 more or 1 less).

(3)⁵⁰% 8

• Step 1: Express (3)⁵⁰ as (3²)²⁵.
• Step 2: (3)² is 9.
• Step 3: 9 divided by 8 leaves a remainder of 1.
• Step 4: (1)²⁵ is 1.

Correct Answer: 1

2⁶⁰ % 17

• Step 1: Express 2⁶⁰ as (2⁴)¹⁵.
• Step 2: (2)⁴ is 16.
• Step 3: 16 divided by 17 leaves a remainder of -1 (or 16).
• Step 4: (-1)¹⁵ is -1.
• Step 5: Convert -1 to a positive remainder by adding the divisor: -1 + 17 = 16.

Correct Answer: 16

7¹⁸ % 24

• Step 1: Express 7¹⁸ as (7²)⁹.
• Step 2: (7)² is 49.
• Step 3: 49 divided by 24 leaves a remainder of 1 (since 24 * 2 = 48).
• Step 4: (1)⁹ is 1.

Correct Answer: 1

(2⁹⁹ × 5⁹⁹)⁹⁹ % 9

• Step 1: Simplify the inside: (2 * 5)⁹⁹ is 10⁹⁹.
• Step 2: The expression is (10⁹⁹)⁹⁹, which is 10^9801 (you
• Step 3: 10 divided by 9 leaves a remainder of 1.
• Step 4: 1⁹⁸⁰¹ is 1.

Correct Answer: 1

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