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Quantitative AptitudeIntermediate4 min study

How to Find the Unit Digit of Large Powers in Seconds

बड़ी घातों का इकाई अंक (Unit Digit) कैसे निकालें: साइक्लिसिटी ट्रिक

Questions asking for the unit digit of expressions like 7²⁰²⁶ or 3⁴⁵ occur regularly in the Number System section of MP Police, MPPSC, and SSC exams. By understanding the concept of cyclicity (repeating unit digit patterns), you can solve these in under 10 seconds without calculating huge powers.

Key Rules & Core Formulas

Understand these foundational properties before solving questions

1Unit Digit Cyclicity Rule

Exponent mod 4 = Remainder determines unit digit

The unit digits of powers of all digits repeat in cycles of at most 4.

Digits 0, 1, 5, 6 always maintain the same unit digit for any positive power (Cyclicity = 1).

2Cyclicity Table

Cycles for 2, 3, 7, 8 (Cycle = 4); 4, 9 (Cycle = 2); 0, 1, 5, 6 (Cycle = 1)

For base ending in 7: 7¹=7, 7²=9, 7³=3, 7⁴=1 (then repeats 7, 9, 3, 1).

If remainder is 0 when dividing power by 4, take the 4th power in the cycle.
Standard Method

Step-by-Step Method to Solve

Follow these universal steps in order to solve any Quantitative Aptitude question accurately:

  1. 1

    Step 1: Focus only on the unit digit of the base

    Ignore all digits of the base except the last (unit) digit. For example, in 347²⁰²⁶, only the base digit 7 matters.

    Pro-Tip: The unit digit of 347²⁰²⁶ is identical to the unit digit of 7²⁰²⁶.
  2. 2

    Step 2: Divide the power (exponent) by 4

    Divide the given power by 4 and note the remainder. To test divisibility by 4, you only need to divide the last two digits of the power.

    Pro-Tip: For power 2026, divide only 26 by 4: 26 ÷ 4 = 6 with Remainder 2.
  3. 3

    Step 3: Replace the huge power with the remainder

    If remainder is 1, take base¹. If remainder is 2, take base². If remainder is 3, take base³. If remainder is 0, take base⁴.

    Pro-Tip: Crucial rule: A remainder of 0 means the 4th power, NOT the 0th power!
  4. 4

    Step 4: Calculate the unit digit of the simplified power

    Compute the unit digit of the small power to get the final answer.

    Pro-Tip: 7² = 49, so the unit digit is 9.

Worked Examples & Walkthroughs

Real questions solved step-by-step showing every calculation

1Example 1Method: Cyclicity Method

Question

Find the unit digit of 7²⁰²⁶.

Step-by-Step Solution:

Step 1: Check base unit digit:Base is 7. The cyclicity of 7 is 4 (powers end in 7, 9, 3, 1).
Step 2: Divide power by 4:Take the last two digits of power 2026, which is 26. 26 ÷ 4 = 6 with a Remainder of 2.
Step 3: Evaluate base with remainder:Since remainder is 2, compute 7².
Step 4: Find unit digit:7² = 49. The unit digit is 9.
Final Result:
The unit digit of 7²⁰²⁶ is 9
Exam Shortcut / Verification: Total time needed: 5 seconds. Just check 26 ÷ 4 = remainder 2 → 7² = 49 → 9.

Common Mistakes to Avoid

Avoid these frequent traps to protect yourself from negative marking

1

Using power 0 when remainder is 0

Why it happens: When a number divides evenly by 4, students think the power becomes 0.
Correct approach: When remainder is 0, always use power 4 (the end of the 4-cycle). For example, 7⁴ = unit digit 1.
Self-Assessment

Try It Yourself (Practice Questions)

Test your understanding on your own first, then reveal the step-by-step solution to check your work:

Practice #1

What is the unit digit of 3⁶⁵?

Quick Revision Summary

  • Only the last digit of the base matters.
  • Divide the last two digits of the power by 4 to get the remainder.
  • If remainder is 1, 2, or 3, use that power. If remainder is 0, use power 4.
  • Digits 0, 1, 5, 6 never change unit digit.
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