Cyclicity: Unit Digit Patterns, Rules and Shortcuts

Cyclicity describes the repeating pattern in the units digit of powers. It helps find the last digit of large powers without calculating the complete value. This page covers standard cycles for digits 0 to 9, exponent-remainder rules, cyclicity shortcuts and examples involving powers, products and expressions.

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What is Cyclicity?

Cyclicity is the repeating sequence of unit digits obtained when a number is raised to successive positive integer powers. The repeated sequence is called the cyclic pattern or cycle of the number.

For a base ending in 2, the unit digits of its powers are 2, 4, 8, 6, and then the pattern repeats. Thus, the cycle length is 4. To find the unit digit of a large power, divide the exponent by the cycle length and use the remainder to select a position in the cycle. A remainder of 0 means the last position of the cycle.

Cyclicity Formula & Tricks

Important Formulas

Exponent remainder rule
r = n mod k

Here, n is the exponent, k is the cycle length and r is the remainder. If r = 0, use the kth digit of the cycle.

Unit digit of a power
For cycle [d₁, d₂, ..., dₖ], aⁿ has unit digit dᵣ when r = n mod k; if r = 0, use dₖ.

The exponent remainder gives the position of the required unit digit in the repeating cycle.

Power of a product
Unit digit of (ab)ⁿ = unit digit of aⁿ × bⁿ

Only the unit digits of a and b are needed. Multiply the resulting unit digits and retain the unit digit of the product.

Quick Tricks

Use only the last digit of the base

The unit digit of a power depends only on the unit digit of its base. Replace the base by its last digit before applying the cycle.

Example: The unit digit of 37²⁵ is the same as the unit digit of 7²⁵.
Treat remainder zero as the last cycle position

If the exponent divides the cycle length exactly, do not use the first digit. Use the final digit in the cycle.

Example: For 2¹², 12 mod 4 = 0, so use the fourth digit of [2, 4, 8, 6], which is 6.
Reduce products before multiplying

For expressions containing several powers, find the unit digit of each factor separately and then multiply only those unit digits.

Example: The unit digit of 3⁴ × 7⁵ is the unit digit of 1 × 7, which is 7.

Cyclicity Concepts

Standard unit digit cycles

The cycle length of a base is determined by its unit digit.

For positive powers, 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]. Digits 0, 1, 5 and 6 have cycle length 1; digits 4 and 9 have cycle length 2; digits 2, 3, 7 and 8 have cycle length 4.

Example: The unit digits of 8¹, 8², 8³ and 8⁴ are 8, 4, 2 and 6. Therefore, the cycle of 8 is [8, 4, 2, 6].

Finding the unit digit of a large power

Divide the exponent by the cycle length and use the remainder as the position in the cycle.

For a base ending in 2, 3, 7 or 8, divide the exponent by 4. For a base ending in 4 or 9, divide it by 2. If the remainder is zero, select the fourth or second cycle position respectively.

Example: To find the unit digit of 7²³, calculate 23 mod 4 = 3. The cycle of 7 is [7, 9, 3, 1], so the answer is 3.

Bases with cycle length one

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

The cycles are [0], [1], [5] and [6]. Therefore, the exponent does not need to be reduced for these last digits.

Example: The unit digit of 145⁹⁹ is 5, and the unit digit of 326⁴⁰ is 6.

Cyclicity in products and expressions

For a product of powers, calculate the unit digit of each factor and multiply the reduced results.

If an expression contains addition or subtraction, first find the unit digit of every term and then perform the operation on those unit digits. For a product, only the final unit digit of each factor is required.

Example: For the unit digit of 2⁷ × 3⁵, 2⁷ has unit digit 8 and 3⁵ has unit digit 3. Since 8 × 3 = 24, the required unit digit is 4.

Cyclicity Video Lessons

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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 Cyclicity Questions

Practise published questions related to this topic.

1If x = √144 ÷ √36 × √49 , then what is the value of x² ?→ 2Which is the largest number that divides 103, 121, 139 and 157 to leave the same remainder in each case?→ 3If a four-digit number 34X1 is divisible by 7, then X2 is:→ 4B = 700 - A. When B is divided by 3, 4 and 5 the remainder is 1 in each case. What is the value of A?→ 5The greatest possible length (in cm) which can be used to measure exactly the lengths 6 metres, 2 metres 70 cm, 11 metres 20 cm is ______.→ 6What is the HCF of 32, 28, and 24?→ 7The 7-digit number 77A444B is divisible by 24. What is the maximum value of (A + B)?→ 8The Highest Common Factor of the two numbers is 12. Which of the following can be the Least Common Multiple of the two numbers?→ 9Which of the following number is NOT divisible by 8?→ 10The sum of a positive number and its square is 90. What is the value of the number?→

Cyclicity Quick Quiz

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Quick Revision Notes

Cyclicity Revision Points

Use these rules to solve unit-digit power questions quickly and accurately.

  • The unit digit of a power depends only on the unit digit of its base.
  • Cycles for 2, 3, 7 and 8 have length 4.
  • Cycles for 4 and 9 have length 2.
  • Digits 0, 1, 5 and 6 have length-1 cycles.
  • Use n mod k to locate the position in a cycle of length k.
  • A remainder of zero means the last position of the cycle.
  • For products, reduce each factor to its unit digit before multiplying.
  • For sums and differences, operate on the unit digits of the separate terms.

Cyclicity FAQs

What is the cyclicity of 2?

The cycle of 2 is [2, 4, 8, 6], so its cycle length is 4.

What is the unit digit of 3²⁰?

The cycle of 3 is [3, 9, 7, 1]. Since 20 mod 4 = 0, use the fourth digit: 1.

What is the unit digit of 9¹⁷?

The cycle of 9 is [9, 1]. Since 17 mod 2 = 1, use the first digit. The unit digit is 9.

What is the unit digit of 4²⁶?

The cycle of 4 is [4, 6]. Since 26 mod 2 = 0, use the second digit. The unit digit is 6.

What is the unit digit of 12¹⁰⁰?

The base ends in 2, whose cycle length is 4. Since 100 mod 4 = 0, use the fourth digit of [2, 4, 8, 6]. The answer is 6.

Does the number of digits in the base affect cyclicity?

No. Only the unit digit of the base affects the unit digit of a positive power. For example, 27⁵ and 7⁵ have the same unit digit.

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