How to Find the Units Digit of Large Exponents on GMAT?

Quick Summary

To find the last digit of large powers on GMAT, keep only the base’s last digit, divide the exponent by 4, and use base^(remainder) to pick the units result. Treat remainder 0 as 4. For subtraction, if the result is negative, finally add 10.

Overview

Questions on finding the last digit of large powers may look impossible at first glance. Numbers like 38¹⁷⁵ or 537³⁹⁵⁸ appear intimidating, but the GMAT never expects you to calculate the entire expression. What matters is the ability to break the problem into small, logical steps. The method rests on two key ideas. First, only the last digit of the base matters. Second, units digits always repeat in cycles of powers of four, so dividing the exponent by 4 is the real shortcut. From there, raising the last digit to the remainder gives the answer. Even special cases, such as when the remainder is 0 or when subtraction leads to a negative digit, can be solved with clear concepts. This kind of reasoning becomes natural only through practice. Building such precision is best achieved with a comprehensive GMAT prep course and sharpened further by taking representative GMAT practice tests under real conditions.

 

Finding the Last Digit of Large Powers in GMAT Quant

When the GMAT asks for the last digit of numbers like 38¹⁷⁵, the question is not testing your ability to multiply endlessly. It is testing your ability to simplify, to see patterns, and to apply logic under time pressure.

The Three-Step Method

1. Take the last digit of the base. The other digits do not matter for the units place.
2. Divide the power by 4 and find the remainder. The cycle of units digits always repeats every 4.
3. Raise the base’s last digit to the remainder. That final value gives the answer.

Example 1: Find the units digit of 38¹⁷⁵

• Last digit of base = 8
• Divide 175 by 4 → remainder 3
• 8³ = 512 → last digit = 2

Answer: 2

Example 2: Find the units digit of 537³⁹⁵⁸

• Last digit of base = 7
• Divide 3958 by 4 → remainder 2
• 7² = 49 → last digit = 9

Answer: 9

The Special Case of Remainder 0

Sometimes the remainder comes out as 0. In such cases, treat it as 4, not 0.

For instance: Find the units digit of 32¹²⁴⁰

• 32¹²⁴⁰ → last digit of base = 2
• Power 1240, when divided by 4 leaves remainder 0
• Replace remainder 0 with 4 → compute 2⁴ = 16
• Last digit=6

Answer: 6

If you had taken 2^0 = 1, you would have gone wrong. This correction ensures consistency.

Another Example: Find the units digit of 518⁷⁸⁰

• Last digit of base = 8
• Power 780, when divided by 4, leaves remainder 0 → replace remainder 0 with remainder 4
• Compute 8^4 = 4096 → last digit = 6

Answer: 6

Handling Negative Results in Subtraction

At times, the question involves subtraction, like 7³ – 2⁴

• 7³ = 343 → last digit 3
• 2⁴ = 16 → last digit 6
• Subtract: 3 – 6 = –3

A negative last digit does not make sense. The way forward is to add 10.

So, –3 + 10 = 7. Final answer: 7.

This borrowing mirrors what you naturally do in arithmetic. The GMAT expects you to remember this adjustment.

Why This Matters

These problems are less about calculation and more about clarity of thought. They reward those who pause, strip away the noise, and focus on the core cycle. The same mindset helps across the exam: spotting patterns, applying logic, and keeping calm under pressure. Developing this instinct takes guided practice. With steady exposure to such problems in a GMAT crash course, you will learn not just to solve them but to solve them quickly and confidently.

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It is entirely possible, and often highly effective, to start your MBA admission consulting journey while preparing for the GMAT. Getting an early start allows us to guide you on both fronts: application planning as well as GMAT preparation. At the very outset, we gather your background details and suggest practical profile-building steps that demand little time but add meaningful substance to your applications. This way, as you focus on the GMAT, you also strengthen your profile and complete much of the application groundwork. With this approach, once your GMAT is behind you, your MBA applications can be finalized within a couple of weeks.

 

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