Greatest Common Divisor (GCD) of 109 and 69
The greatest common divisor (GCD) of 109 and 69 is 1.
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 109 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 | 109 ÷ 69 = 1 remainder 40 |
| 2 | 69 ÷ 40 = 1 remainder 29 |
| 3 | 40 ÷ 29 = 1 remainder 11 |
| 4 | 29 ÷ 11 = 2 remainder 7 |
| 5 | 11 ÷ 7 = 1 remainder 4 |
| 6 | 7 ÷ 4 = 1 remainder 3 |
| 7 | 4 ÷ 3 = 1 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 149 and 160 | 1 |
| 45 and 144 | 9 |
| 117 and 165 | 3 |
| 124 and 87 | 1 |
| 199 and 29 | 1 |