Greatest Common Divisor (GCD) of 106 and 196
The greatest common divisor (GCD) of 106 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 106 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 | 106 ÷ 196 = 0 remainder 106 |
| 2 | 196 ÷ 106 = 1 remainder 90 |
| 3 | 106 ÷ 90 = 1 remainder 16 |
| 4 | 90 ÷ 16 = 5 remainder 10 |
| 5 | 16 ÷ 10 = 1 remainder 6 |
| 6 | 10 ÷ 6 = 1 remainder 4 |
| 7 | 6 ÷ 4 = 1 remainder 2 |
| 8 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 35 and 39 | 1 |
| 112 and 47 | 1 |
| 56 and 164 | 4 |
| 39 and 107 | 1 |
| 104 and 158 | 2 |