How to Find the Remainder of a Large Exponent on the GMAT

Short Answer

To find the remainder of a large exponent, reduce the base and look for a power that leaves +1 or –1 with the divisor. For example, 2³ = 8 leaves remainder +1 with 7, so 2³⁰⁰÷ 7 gives remainder 1.

Overview

Our goal in such questions is to find the remainder of a large power quickly and correctly. In your GMAT preparation course, you learn a simple routine. First reduce the base by the divisor. Then test small powers until one gives remainder +1 or remainder –1. Use that power as the cycle length. Shrink the exponent by this cycle. Replace the expression with 1 or  –1. If the result is  –1, add the divisor to make the remainder positive. Repeat the reduce and shrink steps for any extra factors. This keeps numbers small and avoids heavy multiplication. With steady work on GMAT practice tests, you apply the routine under time limits until it feels natural. The focus stays on clarity, not brute force.

 

Understanding Remainders with Powers

Large powers become simple when you look for a small exponent that makes the base give a remainder of 1 or minus 1 with the divisor. Once you find that exponent, use it as the cycle length to shrink the power and finish in a few clean steps.

An Explanatory Example:

Question: What is the remainder when 9³⁰⁰ is divided by 7?

Remainder of 9 with 7 is 2. Or, 9 mod 7 is 2.

Reduce the base: 9 mod 7 = 2

Test small powers: 2¹ → remainder with 7 is 2
Test small powers: 2² → remainder  with 7 is 4
Test small powers: 2³ → remainder with 7 is 1

Cycle length = 3
Write the exponent: 300 = 3 x 100 + 0
Replace the power: 2³⁰⁰ has the same remainder as 1¹⁰⁰
Final remainder: 1

Same Explanation, In Simple Words…

Take the example of 2³⁰⁰, divided by 7.

Here, 2 itself is not close to +1 or -1 with 7.

But 2 cubed is 8, which leaves a remainder of +1 when divided by 7.

That insight transforms the expression into 8^100 with 7.

Since every 8 leaves a remainder of 1, the answer is simply 1.

The Case of Negative Remainders

Question: What is the remainder of 2³⁰³ with divisor 9?

Here, 2 cubed is 8, which leaves remainder -1 with 9.

Rewriting the problem gives (-1)^101, which is -1.

Since remainders cannot remain negative, we add the divisor 9, giving a final remainder of 8.

This small adjustment is critical and often tested.

Practice Applications

The same approach applies to many other cases. If the base is larger than the divisor, always reduce the base first.

Question: What is the remainder of 92⁶⁰ with divisor 15?

Here, 92⁶⁰ with divisor 15 reduces to 2⁶⁰.

Since 2⁴ is 16, which leaves remainder +1 with 15, the expression simplifies to 1¹⁵.

The required remainder turns out to be 1.

Question: What is the remainder of 2⁶⁰ with divisor 17?

Here, with divisor 17, 2^4 leaves remainder -1,

So, the answer becomes -1^15 = -1,

and after adding 17, the final remainder is 16.

The required remainder turns out to be 16.

Question: What is the remainder of 100¹⁰⁰ with divisor 7?

Here, the base 100 gives remainder 2 with 7.

Thus, 100¹⁰⁰ reduces to 2^100.

Since 2³ = 8 leaves remainder +1 with 7,

the problem further reduces to 1³³ x 2¹,

The required remainder turns out to be 2.

The Larger Lesson

Remainder problems test more than calculations. They test whether you can look past large numbers and recognize patterns. The ability to simplify, to use +1 and -1 intelligently, and to adjust negative remainders correctly reflects the kind of analytical clarity the GMAT seeks. These are not tricks but logical habits that serve well in problem-solving far beyond the test. Developing them not only helps in scoring high but also adds to the structured thinking that will help in your MBA admissions stage and in future professional challenges.

Parting Thought

Remainder problems with large exponents reward a calm, structured approach. You learn to keep only what matters, work with cycles of four, and let checks guide results. Spotting when a power returns 1 or minus 1 trains your eye to see patterns quickly. That habit builds confidence, precision, and speed without strain. Practice the steps until they feel natural: take the last digit, divide the exponent by four, use the remainder, and handle special cases cleanly. Use GMAT simulation to rehearse under timed conditions. With steady practice, tough expressions shrink to a few clear moves, and clarity becomes your default.