Greatest Common Divisor (GCD) of 109 and 137
The greatest common divisor (GCD) of 109 and 137 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 137?
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 ÷ 137 = 0 remainder 109 |
| 2 | 137 ÷ 109 = 1 remainder 28 |
| 3 | 109 ÷ 28 = 3 remainder 25 |
| 4 | 28 ÷ 25 = 1 remainder 3 |
| 5 | 25 ÷ 3 = 8 remainder 1 |
| 6 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 103 and 62 | 1 |
| 74 and 57 | 1 |
| 102 and 95 | 1 |
| 132 and 160 | 4 |
| 122 and 66 | 2 |