Greatest Common Divisor (GCD) of 108 and 66
The greatest common divisor (GCD) of 108 and 66 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 108 and 66?
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 | 108 ÷ 66 = 1 remainder 42 |
| 2 | 66 ÷ 42 = 1 remainder 24 |
| 3 | 42 ÷ 24 = 1 remainder 18 |
| 4 | 24 ÷ 18 = 1 remainder 6 |
| 5 | 18 ÷ 6 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 102 and 88 | 2 |
| 63 and 89 | 1 |
| 184 and 53 | 1 |
| 41 and 96 | 1 |
| 146 and 72 | 2 |