Greatest Common Divisor (GCD) of 78 and 108
The greatest common divisor (GCD) of 78 and 108 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 78 and 108?
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 | 78 ÷ 108 = 0 remainder 78 |
| 2 | 108 ÷ 78 = 1 remainder 30 |
| 3 | 78 ÷ 30 = 2 remainder 18 |
| 4 | 30 ÷ 18 = 1 remainder 12 |
| 5 | 18 ÷ 12 = 1 remainder 6 |
| 6 | 12 ÷ 6 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 126 and 160 | 2 |
| 171 and 65 | 1 |
| 187 and 63 | 1 |
| 182 and 25 | 1 |
| 89 and 44 | 1 |