Greatest Common Divisor (GCD) of 109 and 196
The greatest common divisor (GCD) of 109 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 109 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 | 109 ÷ 196 = 0 remainder 109 |
| 2 | 196 ÷ 109 = 1 remainder 87 |
| 3 | 109 ÷ 87 = 1 remainder 22 |
| 4 | 87 ÷ 22 = 3 remainder 21 |
| 5 | 22 ÷ 21 = 1 remainder 1 |
| 6 | 21 ÷ 1 = 21 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 176 and 126 | 2 |
| 54 and 26 | 2 |
| 106 and 155 | 1 |
| 69 and 173 | 1 |
| 185 and 109 | 1 |