Greatest Common Divisor (GCD) of 107 and 191
The greatest common divisor (GCD) of 107 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 107 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 | 107 ÷ 191 = 0 remainder 107 |
| 2 | 191 ÷ 107 = 1 remainder 84 |
| 3 | 107 ÷ 84 = 1 remainder 23 |
| 4 | 84 ÷ 23 = 3 remainder 15 |
| 5 | 23 ÷ 15 = 1 remainder 8 |
| 6 | 15 ÷ 8 = 1 remainder 7 |
| 7 | 8 ÷ 7 = 1 remainder 1 |
| 8 | 7 ÷ 1 = 7 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 68 and 178 | 2 |
| 189 and 116 | 1 |
| 98 and 57 | 1 |
| 113 and 197 | 1 |
| 91 and 108 | 1 |