Greatest Common Divisor (GCD) of 115 and 180
The greatest common divisor (GCD) of 115 and 180 is 5.
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 115 and 180?
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 | 115 ÷ 180 = 0 remainder 115 |
| 2 | 180 ÷ 115 = 1 remainder 65 |
| 3 | 115 ÷ 65 = 1 remainder 50 |
| 4 | 65 ÷ 50 = 1 remainder 15 |
| 5 | 50 ÷ 15 = 3 remainder 5 |
| 6 | 15 ÷ 5 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 162 and 12 | 6 |
| 199 and 188 | 1 |
| 106 and 173 | 1 |
| 40 and 187 | 1 |
| 112 and 196 | 28 |