Greatest Common Divisor (GCD) of 84 and 129
The greatest common divisor (GCD) of 84 and 129 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 84 and 129?
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 | 84 ÷ 129 = 0 remainder 84 |
| 2 | 129 ÷ 84 = 1 remainder 45 |
| 3 | 84 ÷ 45 = 1 remainder 39 |
| 4 | 45 ÷ 39 = 1 remainder 6 |
| 5 | 39 ÷ 6 = 6 remainder 3 |
| 6 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 190 and 161 | 1 |
| 103 and 152 | 1 |
| 76 and 153 | 1 |
| 88 and 110 | 22 |
| 129 and 31 | 1 |