Greatest Common Divisor (GCD) of 129 and 163
The greatest common divisor (GCD) of 129 and 163 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 163?
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 ÷ 163 = 0 remainder 129 |
| 2 | 163 ÷ 129 = 1 remainder 34 |
| 3 | 129 ÷ 34 = 3 remainder 27 |
| 4 | 34 ÷ 27 = 1 remainder 7 |
| 5 | 27 ÷ 7 = 3 remainder 6 |
| 6 | 7 ÷ 6 = 1 remainder 1 |
| 7 | 6 ÷ 1 = 6 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 62 and 40 | 2 |
| 162 and 43 | 1 |
| 56 and 153 | 1 |
| 172 and 177 | 1 |
| 80 and 114 | 2 |