Greatest Common Divisor (GCD) of 129 and 174
The greatest common divisor (GCD) of 129 and 174 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 174?
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 ÷ 174 = 0 remainder 129 |
| 2 | 174 ÷ 129 = 1 remainder 45 |
| 3 | 129 ÷ 45 = 2 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 |
|---|---|
| 31 and 145 | 1 |
| 167 and 145 | 1 |
| 180 and 32 | 4 |
| 30 and 114 | 6 |
| 73 and 153 | 1 |