Greatest Common Divisor (GCD) of 138 and 76
The greatest common divisor (GCD) of 138 and 76 is 2.
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 138 and 76?
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 | 138 ÷ 76 = 1 remainder 62 |
| 2 | 76 ÷ 62 = 1 remainder 14 |
| 3 | 62 ÷ 14 = 4 remainder 6 |
| 4 | 14 ÷ 6 = 2 remainder 2 |
| 5 | 6 ÷ 2 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 56 and 105 | 7 |
| 64 and 33 | 1 |
| 111 and 139 | 1 |
| 185 and 37 | 37 |
| 125 and 183 | 1 |