Greatest Common Divisor (GCD) of 135 and 189
The greatest common divisor (GCD) of 135 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 135 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 | 135 ÷ 189 = 0 remainder 135 |
| 2 | 189 ÷ 135 = 1 remainder 54 |
| 3 | 135 ÷ 54 = 2 remainder 27 |
| 4 | 54 ÷ 27 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 127 and 17 | 1 |
| 88 and 41 | 1 |
| 102 and 148 | 2 |
| 153 and 162 | 9 |
| 165 and 177 | 3 |