Greatest Common Divisor (GCD) of 186 and 75
The greatest common divisor (GCD) of 186 and 75 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 75?
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 ÷ 75 = 2 remainder 36 |
| 2 | 75 ÷ 36 = 2 remainder 3 |
| 3 | 36 ÷ 3 = 12 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 154 and 44 | 22 |
| 153 and 39 | 3 |
| 121 and 195 | 1 |
| 163 and 139 | 1 |
| 12 and 58 | 2 |