Greatest Common Divisor (GCD) of 109 and 195
The greatest common divisor (GCD) of 109 and 195 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 195?
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 ÷ 195 = 0 remainder 109 |
| 2 | 195 ÷ 109 = 1 remainder 86 |
| 3 | 109 ÷ 86 = 1 remainder 23 |
| 4 | 86 ÷ 23 = 3 remainder 17 |
| 5 | 23 ÷ 17 = 1 remainder 6 |
| 6 | 17 ÷ 6 = 2 remainder 5 |
| 7 | 6 ÷ 5 = 1 remainder 1 |
| 8 | 5 ÷ 1 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 33 and 125 | 1 |
| 112 and 18 | 2 |
| 129 and 175 | 1 |
| 137 and 48 | 1 |
| 42 and 104 | 2 |