Greatest Common Divisor (GCD) of 122 and 196
The greatest common divisor (GCD) of 122 and 196 is 2.
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 122 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 | 122 ÷ 196 = 0 remainder 122 |
| 2 | 196 ÷ 122 = 1 remainder 74 |
| 3 | 122 ÷ 74 = 1 remainder 48 |
| 4 | 74 ÷ 48 = 1 remainder 26 |
| 5 | 48 ÷ 26 = 1 remainder 22 |
| 6 | 26 ÷ 22 = 1 remainder 4 |
| 7 | 22 ÷ 4 = 5 remainder 2 |
| 8 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 35 and 194 | 1 |
| 90 and 56 | 2 |
| 95 and 129 | 1 |
| 196 and 111 | 1 |
| 199 and 33 | 1 |