Greatest Common Divisor (GCD) of 123 and 196
The greatest common divisor (GCD) of 123 and 196 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 196?
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 ÷ 196 = 0 remainder 123 |
| 2 | 196 ÷ 123 = 1 remainder 73 |
| 3 | 123 ÷ 73 = 1 remainder 50 |
| 4 | 73 ÷ 50 = 1 remainder 23 |
| 5 | 50 ÷ 23 = 2 remainder 4 |
| 6 | 23 ÷ 4 = 5 remainder 3 |
| 7 | 4 ÷ 3 = 1 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 122 and 19 | 1 |
| 66 and 108 | 6 |
| 126 and 195 | 3 |
| 16 and 23 | 1 |
| 64 and 94 | 2 |