Greatest Common Divisor (GCD) of 129 and 53
The greatest common divisor (GCD) of 129 and 53 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 53?
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 ÷ 53 = 2 remainder 23 |
| 2 | 53 ÷ 23 = 2 remainder 7 |
| 3 | 23 ÷ 7 = 3 remainder 2 |
| 4 | 7 ÷ 2 = 3 remainder 1 |
| 5 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 188 and 31 | 1 |
| 117 and 200 | 1 |
| 130 and 164 | 2 |
| 123 and 81 | 3 |
| 91 and 94 | 1 |