Greatest Common Divisor (GCD) of 111 and 194
The greatest common divisor (GCD) of 111 and 194 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 111 and 194?
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 | 111 ÷ 194 = 0 remainder 111 |
| 2 | 194 ÷ 111 = 1 remainder 83 |
| 3 | 111 ÷ 83 = 1 remainder 28 |
| 4 | 83 ÷ 28 = 2 remainder 27 |
| 5 | 28 ÷ 27 = 1 remainder 1 |
| 6 | 27 ÷ 1 = 27 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 110 and 137 | 1 |
| 185 and 159 | 1 |
| 27 and 119 | 1 |
| 140 and 54 | 2 |
| 162 and 148 | 2 |