Greatest Common Divisor (GCD) of 114 and 69
The greatest common divisor (GCD) of 114 and 69 is 3.
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 114 and 69?
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 | 114 ÷ 69 = 1 remainder 45 |
| 2 | 69 ÷ 45 = 1 remainder 24 |
| 3 | 45 ÷ 24 = 1 remainder 21 |
| 4 | 24 ÷ 21 = 1 remainder 3 |
| 5 | 21 ÷ 3 = 7 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 139 and 32 | 1 |
| 40 and 78 | 2 |
| 182 and 18 | 2 |
| 180 and 106 | 2 |
| 192 and 130 | 2 |