Greatest Common Divisor (GCD) of 108 and 187
The greatest common divisor (GCD) of 108 and 187 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 108 and 187?
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 ÷ 187 = 0 remainder 108 |
| 2 | 187 ÷ 108 = 1 remainder 79 |
| 3 | 108 ÷ 79 = 1 remainder 29 |
| 4 | 79 ÷ 29 = 2 remainder 21 |
| 5 | 29 ÷ 21 = 1 remainder 8 |
| 6 | 21 ÷ 8 = 2 remainder 5 |
| 7 | 8 ÷ 5 = 1 remainder 3 |
| 8 | 5 ÷ 3 = 1 remainder 2 |
| 9 | 3 ÷ 2 = 1 remainder 1 |
| 10 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 12 and 14 | 2 |
| 53 and 149 | 1 |
| 154 and 75 | 1 |
| 174 and 101 | 1 |
| 33 and 83 | 1 |