Greatest Common Divisor (GCD) of 186 and 111
The greatest common divisor (GCD) of 186 and 111 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 186 and 111?
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 | 186 ÷ 111 = 1 remainder 75 |
| 2 | 111 ÷ 75 = 1 remainder 36 |
| 3 | 75 ÷ 36 = 2 remainder 3 |
| 4 | 36 ÷ 3 = 12 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 134 and 171 | 1 |
| 29 and 128 | 1 |
| 13 and 72 | 1 |
| 42 and 162 | 6 |
| 152 and 30 | 2 |