Concept:
Unit digit: It is defined as the digit at once place of a given number. To calculate the unit digit, of a certain number with some power or product of numbers, we only have to focus on the digit at the one place of numbers.
Cyclicity: The concept of cyclicity is used to identify the last digit of the number. The cyclicity of any number is about the last digit and how they appear in a certain defined manner.
Procedure to find unit digit:
Step-1: Take the given power like the power of the unit digit of a given number.
Step-2: Divide the power by cyclicity of a given number
If it is completely divisible, then the unit digit will be digit at once place to the power its cyclicity.
If it is not divisible completely, replace the power with the remainder obtained, let xy (where x is a unit digit of a given number and y is remainder).
Step-3: Now, the unit digit will be obtained by simplifying xy.
Calculation:
12123
Unit digit of the given number (base) is 3, and its cyclicity 4. Hence,
Step-1 123/4 ⇒ Remainder = 3
Step-2 (2)3 = 8
Unit digit of 12123 will be 8
Alternate Method
The last digit series of powers of 12 is 2, 4, 8, 6, 2.
∴ Numbers of form 124k + 1 : 2
∴ Numbers of form 124k + 2 : 4
∴ Numbers of form 124k + 3 : 8
∴ Numbers of form 124k + 4 : 6
∴ 12123 = 124 × 30 + 3 will have 8 as unit digit.
Additional Information
| Number (n) | Unit digit of n1 | Unit digit of n2 | Unit digit of n3 | Unit digit of n4 | Cyclicity |
|---|
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 4 | 8 | 6 | 4 |
| 3 | 3 | 9 | 7 | 1 | 4 |
| 4 | 4 | 6 | 4 | 6 | 2 |
| 5 | 5 | 5 | 5 | 5 | 1 |
| 6 | 6 | 6 | 6 | 6 | 1 |
| 7 | 7 | 9 | 3 | 1 | 4 |
| 8 | 8 | 4 | 2 | 6 | 4 |
| 9 | 9 | 1 | 9 | 1 | 2 |