Unit Digit: Patterns, Formulas and Shortcut Methods

Unit Digit is the digit in the ones place of a number. It can be found by observing the last digit of powers, products and sums. This page explains unit digit patterns, cyclicity, special cases and calculation shortcuts for solving numerical expressions quickly and accurately.

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What Is a Unit Digit?

The unit digit is the rightmost digit of a number, also called its ones digit. For example, the unit digit of 7,438 is 8.

Only the last digit is required when finding a unit digit. For a power, the last digits usually repeat in a cycle. For example, the unit digits of powers of 2 are 2, 4, 8, 6, and then the pattern repeats. Therefore, the unit digit of 2^10 is 4 because 10 divided by the cycle length 4 leaves remainder 2.

Unit Digit Formula & Tricks

Important Formulas

Unit digit of a product
Unit digit of (a × b) = (unit digit of a × unit digit of b) mod 10

Multiply only the last digits of the factors and retain the last digit of the result. For example, the unit digit of 47 × 83 is the unit digit of 7 × 3 = 21, which is 1.

Unit digit of a sum
Unit digit of (a + b) = (unit digit of a + unit digit of b) mod 10

Add the last digits and retain the ones digit. For example, the unit digit of 238 + 457 is the unit digit of 8 + 7 = 15, which is 5.

Power cycle rule
If a cycle has length L, use exponent remainder r = n mod L; if r = 0, use the Lth term

Find the repeating cycle of the last digit, divide the exponent by the cycle length and select the term indicated by the remainder.

Modular form
Unit digit of N = N mod 10

The remainder obtained when a number is divided by 10 gives its unit digit. For example, 586 mod 10 = 6.

Quick Tricks

Use only the last digit of each factor

For multiplication, digits before the unit place do not affect the final digit. Replace every factor by its unit digit before multiplying.

Example: The unit digit of 124 × 357 × 86 is the unit digit of 4 × 7 × 6 = 168, so the answer is 8.
Use the exponent cycle instead of calculating the power

For bases ending in 2, 3, 7 or 8, the unit digit cycle has length 4. Reduce the exponent modulo 4 and select the corresponding term.

Example: For 7^23, 23 mod 4 = 3. The cycle of 7 is 7, 9, 3, 1, so the unit digit is 3.
Handle remainder zero as the last cycle term

When the exponent is exactly divisible by the cycle length, use the final term of the cycle rather than the zeroth term.

Example: For 3^20, 20 mod 4 = 0. The fourth term of the cycle 3, 9, 7, 1 is 1, so the unit digit is 1.
Check special last digits first

Powers of numbers ending in 0, 1, 5 or 6 always end in the same digit.

Example: The unit digit of 35^18 is 5, and the unit digit of 126^99 is 6.

Unit Digit Concepts

Unit Digit Cycles of Powers

The unit digit of a power follows a repeating cycle determined by the base's last digit.

The standard cycles are: 0 → 0; 1 → 1; 2 → 2, 4, 8, 6; 3 → 3, 9, 7, 1; 4 → 4, 6; 5 → 5; 6 → 6; 7 → 7, 9, 3, 1; 8 → 8, 4, 2, 6; and 9 → 9, 1. The cycle length is 1, 2 or 4.

Example: The unit digits of powers of 8 are 8, 4, 2, 6. Since 8^15 has 15 mod 4 = 3, its unit digit is the third term, 2.

Finding the Unit Digit of a Power

To find the unit digit of a^n, use the cycle of the last digit of a and reduce n by the cycle length.

First identify the last digit of the base. Next, divide the exponent by the cycle length. A remainder of 1, 2, 3 or 4 selects the corresponding cycle term; when the remainder is 0, select the final term. For cycles of length 2, use remainder 1 or 2.

Example: For 4^37, the cycle is 4, 6. Since 37 mod 2 = 1, the unit digit is 4.

Special Cases Ending in 0, 1, 5 and 6

Any positive power of a number ending in 0, 1, 5 or 6 has the same unit digit as its base.

Multiplication preserves these last digits: a number ending in 0 always produces 0, one ending in 1 produces 1, one ending in 5 produces 5, and one ending in 6 produces 6 when raised to any positive integer power.

Example: The unit digit of 91^24 is 1, while the unit digit of 45^12 is 5.

Unit Digit of Products and Quotients

For a product, multiply the unit digits of the factors and retain the last digit of the result.

If the expression contains powers, find the unit digit of each power first, then multiply those digits modulo 10. Direct division is not generally valid for unit-digit questions because division may not produce an integer or may require additional divisibility information.

Example: The unit digit of 2^6 × 3^4 is found from 2^6 → 4 and 3^4 → 1. Thus, the unit digit is 4 × 1 = 4.

Unit Digit of Sums and Differences

For a sum or difference, operate on the unit digits and retain the resulting ones digit.

For addition, add the last digits; for subtraction, subtract the last digits and interpret the result modulo 10. If the result is negative, add 10 to obtain the equivalent unit digit.

Example: The unit digit of 738 − 465 is the unit digit of 8 − 5 = 3. The unit digit of 402 − 589 is the unit digit of −7, which is 3 because −7 mod 10 = 3.

Unit Digit Video Lessons

Watch short topic-wise lessons for quick revision.

14 Lessons
Lesson 1 of 14 Quick Revision

Unit Digit and Cyclicity of Powers

Learn how to determine the unit digit of powers using repeating cyclic patterns, with methods applicable to number system problems in competitive examinations.

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Practice Unit Digit Questions

Practise published questions related to this topic.

1Find the HCF of the numbers 2520, 3600, 5880.→ 2Find the smallest digit x such that the 6-digit number 28x219 is divisible by both 9 and 11.→ 3If the number 2X73Y5 is divisible by 11, then find the value of (X-Y).→ 4If √210.25 = 14.5, then what is the value of √0.00021025?→ 5Which of the following number is neither divisible by 3 nor by 4?→ 6Find the HCF of 2.8, 11.2 and 0.84.→ 7Two positive numbers are in the ratio 2:3. If the product of their LCM and HCF is 294, then find the sum of the two numbers.→ 8√(0.0225/0.04) = ?→ 9The difference between the two numbers is 1365. On dividing the larger by the smaller, the quotient is 6, and the remainder is 15. Find the smaller number.→ 10Find the greatest number that will divide 186, 321 and 402 so as to leave the same remainder in each case.→

Unit Digit Quick Quiz

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Quick Revision Notes

Unit Digit Quick Revision

Use these rules to solve unit digit questions involving powers, products, sums and differences.

  • The unit digit of N is N mod 10.
  • For multiplication, retain only the unit digit of each factor.
  • Power cycles have lengths 1, 2 or 4 for decimal last digits.
  • For cycle length L, use n mod L; if the remainder is 0, select the Lth cycle term.
  • The cycles of 2, 3, 7 and 8 have length 4.
  • The cycles of 4 and 9 have length 2.
  • Positive powers of numbers ending in 0, 1, 5 or 6 retain the same unit digit.
  • For sums and differences, calculate with last digits and retain the result modulo 10.

Unit Digit FAQs

What is the unit digit of 2^100?

The cycle of 2 is 2, 4, 8, 6. Since 100 mod 4 = 0, use the fourth term. The unit digit is 6.

What is the unit digit of 9^47?

The cycle of 9 is 9, 1. Since 47 mod 2 = 1, the unit digit is 9.

What is the unit digit of 6^125?

Every positive power of a number ending in 6 ends in 6. Therefore, the unit digit is 6.

How do you find the unit digit of 23^14 × 17^9?

The unit digit of 3^14 is 9 because 14 mod 4 = 2, and the unit digit of 7^9 is 7 because 9 mod 4 = 1. Their product ends in 9 × 7 = 63, so the answer is 3.

What is the unit digit of 4^18 + 7^11?

The cycle of 4 gives 4^18 → 4, and the cycle of 7 gives 7^11 → 7. Their sum is 11, so the unit digit is 1.

Why is the unit digit of 5^0 different from positive powers of 5?

For every non-zero number, a^0 = 1. Thus, 5^0 has unit digit 1, whereas every positive power of 5 has unit digit 5.

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