Greatest Common Divisor (GCD) of 69 and 12
The greatest common divisor (GCD) of 69 and 12 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 12?
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 ÷ 12 = 5 remainder 9 |
| 2 | 12 ÷ 9 = 1 remainder 3 |
| 3 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 172 and 172 | 172 |
| 29 and 97 | 1 |
| 199 and 71 | 1 |
| 190 and 184 | 2 |
| 106 and 118 | 2 |