Greatest Common Divisor (GCD) of 121 and 196
The greatest common divisor (GCD) of 121 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 121 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 | 121 ÷ 196 = 0 remainder 121 |
| 2 | 196 ÷ 121 = 1 remainder 75 |
| 3 | 121 ÷ 75 = 1 remainder 46 |
| 4 | 75 ÷ 46 = 1 remainder 29 |
| 5 | 46 ÷ 29 = 1 remainder 17 |
| 6 | 29 ÷ 17 = 1 remainder 12 |
| 7 | 17 ÷ 12 = 1 remainder 5 |
| 8 | 12 ÷ 5 = 2 remainder 2 |
| 9 | 5 ÷ 2 = 2 remainder 1 |
| 10 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 45 and 121 | 1 |
| 196 and 19 | 1 |
| 186 and 72 | 6 |
| 140 and 166 | 2 |
| 158 and 128 | 2 |