Greatest Common Divisor (GCD) of 193 and 119
The greatest common divisor (GCD) of 193 and 119 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 193 and 119?
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 | 193 ÷ 119 = 1 remainder 74 |
| 2 | 119 ÷ 74 = 1 remainder 45 |
| 3 | 74 ÷ 45 = 1 remainder 29 |
| 4 | 45 ÷ 29 = 1 remainder 16 |
| 5 | 29 ÷ 16 = 1 remainder 13 |
| 6 | 16 ÷ 13 = 1 remainder 3 |
| 7 | 13 ÷ 3 = 4 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 184 and 68 | 4 |
| 46 and 71 | 1 |
| 127 and 139 | 1 |
| 15 and 188 | 1 |
| 73 and 137 | 1 |