Greatest Common Divisor (GCD) of 125 and 109
The greatest common divisor (GCD) of 125 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 125 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 | 125 ÷ 109 = 1 remainder 16 |
| 2 | 109 ÷ 16 = 6 remainder 13 |
| 3 | 16 ÷ 13 = 1 remainder 3 |
| 4 | 13 ÷ 3 = 4 remainder 1 |
| 5 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 40 and 107 | 1 |
| 189 and 87 | 3 |
| 148 and 32 | 4 |
| 149 and 109 | 1 |
| 142 and 91 | 1 |