Greatest Common Divisor (GCD) of 69 and 138
The greatest common divisor (GCD) of 69 and 138 is 69.
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 69 and 138?
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 | 69 ÷ 138 = 0 remainder 69 |
| 2 | 138 ÷ 69 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 146 and 172 | 2 |
| 10 and 167 | 1 |
| 138 and 25 | 1 |
| 161 and 54 | 1 |
| 45 and 130 | 5 |