Greatest Common Divisor (GCD) of 54 and 186
The greatest common divisor (GCD) of 54 and 186 is 6.
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 54 and 186?
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 | 54 ÷ 186 = 0 remainder 54 |
| 2 | 186 ÷ 54 = 3 remainder 24 |
| 3 | 54 ÷ 24 = 2 remainder 6 |
| 4 | 24 ÷ 6 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 87 and 109 | 1 |
| 102 and 178 | 2 |
| 159 and 120 | 3 |
| 123 and 96 | 3 |
| 166 and 21 | 1 |