Greatest Common Divisor (GCD) of 108 and 189
The greatest common divisor (GCD) of 108 and 189 is 27.
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 108 and 189?
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 | 108 ÷ 189 = 0 remainder 108 |
| 2 | 189 ÷ 108 = 1 remainder 81 |
| 3 | 108 ÷ 81 = 1 remainder 27 |
| 4 | 81 ÷ 27 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 64 and 139 | 1 |
| 135 and 38 | 1 |
| 61 and 115 | 1 |
| 188 and 23 | 1 |
| 114 and 83 | 1 |