Greatest Common Divisor (GCD) of 129 and 186
The greatest common divisor (GCD) of 129 and 186 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 186?
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 ÷ 186 = 0 remainder 129 |
| 2 | 186 ÷ 129 = 1 remainder 57 |
| 3 | 129 ÷ 57 = 2 remainder 15 |
| 4 | 57 ÷ 15 = 3 remainder 12 |
| 5 | 15 ÷ 12 = 1 remainder 3 |
| 6 | 12 ÷ 3 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 126 and 176 | 2 |
| 29 and 75 | 1 |
| 151 and 23 | 1 |
| 18 and 128 | 2 |
| 188 and 58 | 2 |