Greatest Common Divisor (GCD) of 123 and 194
The greatest common divisor (GCD) of 123 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 123 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 | 123 ÷ 194 = 0 remainder 123 |
| 2 | 194 ÷ 123 = 1 remainder 71 |
| 3 | 123 ÷ 71 = 1 remainder 52 |
| 4 | 71 ÷ 52 = 1 remainder 19 |
| 5 | 52 ÷ 19 = 2 remainder 14 |
| 6 | 19 ÷ 14 = 1 remainder 5 |
| 7 | 14 ÷ 5 = 2 remainder 4 |
| 8 | 5 ÷ 4 = 1 remainder 1 |
| 9 | 4 ÷ 1 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 190 and 188 | 2 |
| 129 and 189 | 3 |
| 122 and 16 | 2 |
| 139 and 30 | 1 |
| 23 and 110 | 1 |