Greatest Common Divisor (GCD) of 59 and 108
The greatest common divisor (GCD) of 59 and 108 is 1.
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 59 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 | 59 ÷ 108 = 0 remainder 59 |
| 2 | 108 ÷ 59 = 1 remainder 49 |
| 3 | 59 ÷ 49 = 1 remainder 10 |
| 4 | 49 ÷ 10 = 4 remainder 9 |
| 5 | 10 ÷ 9 = 1 remainder 1 |
| 6 | 9 ÷ 1 = 9 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 126 and 39 | 3 |
| 198 and 184 | 2 |
| 89 and 56 | 1 |
| 182 and 178 | 2 |
| 180 and 25 | 5 |