Greatest Common Divisor (GCD) of 129 and 77
The greatest common divisor (GCD) of 129 and 77 is 1.
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 77?
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 ÷ 77 = 1 remainder 52 |
| 2 | 77 ÷ 52 = 1 remainder 25 |
| 3 | 52 ÷ 25 = 2 remainder 2 |
| 4 | 25 ÷ 2 = 12 remainder 1 |
| 5 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 148 and 148 | 148 |
| 179 and 200 | 1 |
| 167 and 121 | 1 |
| 58 and 96 | 2 |
| 178 and 140 | 2 |