Greatest Common Divisor (GCD) of 126 and 196
The greatest common divisor (GCD) of 126 and 196 is 14.
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 126 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 | 126 ÷ 196 = 0 remainder 126 |
| 2 | 196 ÷ 126 = 1 remainder 70 |
| 3 | 126 ÷ 70 = 1 remainder 56 |
| 4 | 70 ÷ 56 = 1 remainder 14 |
| 5 | 56 ÷ 14 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 34 and 138 | 2 |
| 169 and 98 | 1 |
| 90 and 161 | 1 |
| 183 and 157 | 1 |
| 105 and 194 | 1 |