Greatest Common Divisor (GCD) of 52 and 138
The greatest common divisor (GCD) of 52 and 138 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 52 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 | 52 ÷ 138 = 0 remainder 52 |
| 2 | 138 ÷ 52 = 2 remainder 34 |
| 3 | 52 ÷ 34 = 1 remainder 18 |
| 4 | 34 ÷ 18 = 1 remainder 16 |
| 5 | 18 ÷ 16 = 1 remainder 2 |
| 6 | 16 ÷ 2 = 8 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 14 and 132 | 2 |
| 111 and 129 | 3 |
| 108 and 72 | 36 |
| 163 and 54 | 1 |
| 178 and 137 | 1 |