Greatest Common Divisor (GCD) of 69 and 114
The greatest common divisor (GCD) of 69 and 114 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 69 and 114?
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 ÷ 114 = 0 remainder 69 |
| 2 | 114 ÷ 69 = 1 remainder 45 |
| 3 | 69 ÷ 45 = 1 remainder 24 |
| 4 | 45 ÷ 24 = 1 remainder 21 |
| 5 | 24 ÷ 21 = 1 remainder 3 |
| 6 | 21 ÷ 3 = 7 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 176 and 89 | 1 |
| 39 and 170 | 1 |
| 191 and 178 | 1 |
| 146 and 80 | 2 |
| 51 and 188 | 1 |