Greatest Common Divisor (GCD) of 129 and 138
The greatest common divisor (GCD) of 129 and 138 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 129 and 138?
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 | 129 ÷ 138 = 0 remainder 129 |
| 2 | 138 ÷ 129 = 1 remainder 9 |
| 3 | 129 ÷ 9 = 14 remainder 3 |
| 4 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 195 and 85 | 5 |
| 140 and 75 | 5 |
| 39 and 163 | 1 |
| 167 and 69 | 1 |
| 107 and 184 | 1 |