Greatest Common Divisor (GCD) of 109 and 191
The greatest common divisor (GCD) of 109 and 191 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 191?
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 ÷ 191 = 0 remainder 109 |
| 2 | 191 ÷ 109 = 1 remainder 82 |
| 3 | 109 ÷ 82 = 1 remainder 27 |
| 4 | 82 ÷ 27 = 3 remainder 1 |
| 5 | 27 ÷ 1 = 27 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 194 and 32 | 2 |
| 112 and 79 | 1 |
| 121 and 182 | 1 |
| 126 and 82 | 2 |
| 113 and 120 | 1 |