Greatest Common Divisor (GCD) of 117 and 193
The greatest common divisor (GCD) of 117 and 193 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 193?
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 ÷ 193 = 0 remainder 117 |
| 2 | 193 ÷ 117 = 1 remainder 76 |
| 3 | 117 ÷ 76 = 1 remainder 41 |
| 4 | 76 ÷ 41 = 1 remainder 35 |
| 5 | 41 ÷ 35 = 1 remainder 6 |
| 6 | 35 ÷ 6 = 5 remainder 5 |
| 7 | 6 ÷ 5 = 1 remainder 1 |
| 8 | 5 ÷ 1 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 68 and 60 | 4 |
| 132 and 63 | 3 |
| 147 and 109 | 1 |
| 48 and 23 | 1 |
| 37 and 149 | 1 |