Greatest Common Divisor (GCD) of 117 and 191
The greatest common divisor (GCD) of 117 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 117 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 | 117 ÷ 191 = 0 remainder 117 |
| 2 | 191 ÷ 117 = 1 remainder 74 |
| 3 | 117 ÷ 74 = 1 remainder 43 |
| 4 | 74 ÷ 43 = 1 remainder 31 |
| 5 | 43 ÷ 31 = 1 remainder 12 |
| 6 | 31 ÷ 12 = 2 remainder 7 |
| 7 | 12 ÷ 7 = 1 remainder 5 |
| 8 | 7 ÷ 5 = 1 remainder 2 |
| 9 | 5 ÷ 2 = 2 remainder 1 |
| 10 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 111 and 196 | 1 |
| 157 and 146 | 1 |
| 117 and 14 | 1 |
| 22 and 11 | 11 |
| 56 and 85 | 1 |