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What is Cyclic Power?

Published in Number Theory 3 mins read

Cyclic power, also known as cyclicity in number systems, is a technique to predict the unit digits of a number raised to a power (ab). In simpler terms, it's about finding patterns in the last digit when a number is multiplied by itself repeatedly.

Understanding the Concept

The power concept or the cyclicity of a number, as stated in the reference, focuses on the repeating patterns of unit digits when a number is raised to consecutive powers. Let's explore this with examples.

Examples of Cyclic Power

Example 1: Cyclicity of 2

Let's observe the unit digits of powers of 2:

  • 21 = 2
  • 22 = 4
  • 23 = 8
  • 24 = 16
  • 25 = 32
  • 26 = 64

Notice the unit digits repeat in a cycle: 2, 4, 8, 6, 2, 4,... The cyclicity of 2 is 4, meaning the pattern of the unit digit repeats every four powers.

Example 2: Cyclicity of 3

  • 31 = 3
  • 32 = 9
  • 33 = 27
  • 34 = 81
  • 35 = 243

The cycle of unit digits is 3, 9, 7, 1. The cyclicity of 3 is also 4.

How to Find the Unit Digit Using Cyclic Power

To find the unit digit of a number ab using cyclic power:

  1. Identify the Base (a): Determine the base number whose cyclicity you need to consider.
  2. Find the Cyclicity: Determine the length of the repeating pattern of the unit digits of the base raised to successive powers.
  3. Divide the Exponent (b) by the Cyclicity: Divide the exponent b by the cyclicity found in the previous step.
  4. Find the Remainder: Note the remainder from the division in step 3.
  5. Determine the Unit Digit: The unit digit of ab is the same as the unit digit of a raised to the power of the remainder found in step 4. If the remainder is 0, the unit digit is the same as the unit digit of a raised to the power of the cyclicity.

Example Calculation

Find the unit digit of 725.

  1. Base: 7
  2. Cyclicity of 7: The unit digits of powers of 7 are 7, 9, 3, 1. The cyclicity is 4.
  3. Divide Exponent: 25 รท 4 = 6 with a remainder of 1.
  4. Remainder: 1
  5. Unit Digit: The unit digit of 725 is the same as the unit digit of 71, which is 7.

Practical Application

Cyclic power is a useful concept in:

  • Number Theory: It simplifies problems involving large exponents.
  • Competitive Exams: Helps in quickly solving problems related to finding unit digits.
  • Cryptography: Certain cryptographic algorithms utilize properties of modular arithmetic, where cyclic patterns can play a role.

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