Greatest Common Divisor (GCD) of 123 and 198
The greatest common divisor (GCD) of 123 and 198 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 123 and 198?
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 | 123 ÷ 198 = 0 remainder 123 |
| 2 | 198 ÷ 123 = 1 remainder 75 |
| 3 | 123 ÷ 75 = 1 remainder 48 |
| 4 | 75 ÷ 48 = 1 remainder 27 |
| 5 | 48 ÷ 27 = 1 remainder 21 |
| 6 | 27 ÷ 21 = 1 remainder 6 |
| 7 | 21 ÷ 6 = 3 remainder 3 |
| 8 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 124 and 117 | 1 |
| 186 and 160 | 2 |
| 179 and 49 | 1 |
| 62 and 174 | 2 |
| 181 and 87 | 1 |