Greatest Common Divisor (GCD) of 69 and 132
The greatest common divisor (GCD) of 69 and 132 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 132?
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 ÷ 132 = 0 remainder 69 |
| 2 | 132 ÷ 69 = 1 remainder 63 |
| 3 | 69 ÷ 63 = 1 remainder 6 |
| 4 | 63 ÷ 6 = 10 remainder 3 |
| 5 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 101 and 28 | 1 |
| 179 and 54 | 1 |
| 113 and 49 | 1 |
| 194 and 192 | 2 |
| 149 and 165 | 1 |