Greatest Common Divisor (GCD) of 111 and 189
The greatest common divisor (GCD) of 111 and 189 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 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 | 111 ÷ 189 = 0 remainder 111 |
| 2 | 189 ÷ 111 = 1 remainder 78 |
| 3 | 111 ÷ 78 = 1 remainder 33 |
| 4 | 78 ÷ 33 = 2 remainder 12 |
| 5 | 33 ÷ 12 = 2 remainder 9 |
| 6 | 12 ÷ 9 = 1 remainder 3 |
| 7 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 119 and 125 | 1 |
| 185 and 103 | 1 |
| 131 and 74 | 1 |
| 131 and 50 | 1 |
| 177 and 40 | 1 |