Greatest Common Divisor (GCD) of 189 and 123
The greatest common divisor (GCD) of 189 and 123 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 189 and 123?
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 | 189 ÷ 123 = 1 remainder 66 |
| 2 | 123 ÷ 66 = 1 remainder 57 |
| 3 | 66 ÷ 57 = 1 remainder 9 |
| 4 | 57 ÷ 9 = 6 remainder 3 |
| 5 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 137 and 198 | 1 |
| 55 and 35 | 5 |
| 59 and 136 | 1 |
| 41 and 129 | 1 |
| 160 and 77 | 1 |