Greatest Common Divisor (GCD) of 111 and 198
The greatest common divisor (GCD) of 111 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 111 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 | 111 ÷ 198 = 0 remainder 111 |
| 2 | 198 ÷ 111 = 1 remainder 87 |
| 3 | 111 ÷ 87 = 1 remainder 24 |
| 4 | 87 ÷ 24 = 3 remainder 15 |
| 5 | 24 ÷ 15 = 1 remainder 9 |
| 6 | 15 ÷ 9 = 1 remainder 6 |
| 7 | 9 ÷ 6 = 1 remainder 3 |
| 8 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 67 and 33 | 1 |
| 123 and 73 | 1 |
| 28 and 83 | 1 |
| 98 and 193 | 1 |
| 178 and 165 | 1 |