Greatest Common Divisor (GCD) of 123 and 138
The greatest common divisor (GCD) of 123 and 138 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 123 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 | 123 ÷ 138 = 0 remainder 123 |
| 2 | 138 ÷ 123 = 1 remainder 15 |
| 3 | 123 ÷ 15 = 8 remainder 3 |
| 4 | 15 ÷ 3 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 163 and 117 | 1 |
| 182 and 42 | 14 |
| 195 and 169 | 13 |
| 42 and 64 | 2 |
| 46 and 141 | 1 |