Greatest Common Divisor (GCD) of 69 and 115
The greatest common divisor (GCD) of 69 and 115 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 69 and 115?
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 | 69 ÷ 115 = 0 remainder 69 |
| 2 | 115 ÷ 69 = 1 remainder 46 |
| 3 | 69 ÷ 46 = 1 remainder 23 |
| 4 | 46 ÷ 23 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 64 and 194 | 2 |
| 21 and 200 | 1 |
| 148 and 124 | 4 |
| 95 and 32 | 1 |
| 193 and 83 | 1 |