Greatest Common Divisor (GCD) of 67 and 109
The greatest common divisor (GCD) of 67 and 109 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 67 and 109?
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 | 67 ÷ 109 = 0 remainder 67 |
| 2 | 109 ÷ 67 = 1 remainder 42 |
| 3 | 67 ÷ 42 = 1 remainder 25 |
| 4 | 42 ÷ 25 = 1 remainder 17 |
| 5 | 25 ÷ 17 = 1 remainder 8 |
| 6 | 17 ÷ 8 = 2 remainder 1 |
| 7 | 8 ÷ 1 = 8 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 24 and 129 | 3 |
| 190 and 39 | 1 |
| 65 and 143 | 13 |
| 66 and 43 | 1 |
| 46 and 37 | 1 |