Greatest Common Divisor (GCD) of 109 and 75
The greatest common divisor (GCD) of 109 and 75 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 75?
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 ÷ 75 = 1 remainder 34 |
| 2 | 75 ÷ 34 = 2 remainder 7 |
| 3 | 34 ÷ 7 = 4 remainder 6 |
| 4 | 7 ÷ 6 = 1 remainder 1 |
| 5 | 6 ÷ 1 = 6 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 135 and 123 | 3 |
| 130 and 162 | 2 |
| 136 and 191 | 1 |
| 124 and 171 | 1 |
| 129 and 119 | 1 |