Greatest Common Divisor (GCD) of 81 and 123
The greatest common divisor (GCD) of 81 and 123 is 3.
What is the Greatest Common Divisor (GCD)?
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is useful in simplifying fractions, finding common factors, and in number theory.
How to Calculate the GCD of 81 and 123?
We use the Euclidean algorithm, which involves the following steps:
- Divide the larger number by the smaller number.
- Replace the larger number with the smaller number and the smaller number with the remainder from the division.
- Repeat this process until the remainder is zero.
- The non-zero remainder just before zero is the GCD.
Step-by-Step Euclidean Algorithm
| Step | Calculation |
|---|---|
| 1 | 81 ÷ 123 = 0 remainder 81 |
| 2 | 123 ÷ 81 = 1 remainder 42 |
| 3 | 81 ÷ 42 = 1 remainder 39 |
| 4 | 42 ÷ 39 = 1 remainder 3 |
| 5 | 39 ÷ 3 = 13 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 187 and 84 | 1 |
| 113 and 90 | 1 |
| 196 and 122 | 2 |
| 185 and 116 | 1 |
| 90 and 166 | 2 |