Greatest Common Divisor (GCD) of 109 and 193
The greatest common divisor (GCD) of 109 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 109 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 | 109 ÷ 193 = 0 remainder 109 |
| 2 | 193 ÷ 109 = 1 remainder 84 |
| 3 | 109 ÷ 84 = 1 remainder 25 |
| 4 | 84 ÷ 25 = 3 remainder 9 |
| 5 | 25 ÷ 9 = 2 remainder 7 |
| 6 | 9 ÷ 7 = 1 remainder 2 |
| 7 | 7 ÷ 2 = 3 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 88 and 141 | 1 |
| 115 and 98 | 1 |
| 158 and 55 | 1 |
| 11 and 31 | 1 |
| 38 and 119 | 1 |