Greatest Common Divisor (GCD) of 115 and 184
The greatest common divisor (GCD) of 115 and 184 is 23.
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 184?
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 ÷ 184 = 0 remainder 115 |
| 2 | 184 ÷ 115 = 1 remainder 69 |
| 3 | 115 ÷ 69 = 1 remainder 46 |
| 4 | 69 ÷ 46 = 1 remainder 23 |
| 5 | 46 ÷ 23 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 109 and 104 | 1 |
| 43 and 192 | 1 |
| 15 and 172 | 1 |
| 85 and 92 | 1 |
| 51 and 20 | 1 |