Greatest Common Divisor (GCD) of 84 and 106
The greatest common divisor (GCD) of 84 and 106 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 84 and 106?
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 | 84 ÷ 106 = 0 remainder 84 |
| 2 | 106 ÷ 84 = 1 remainder 22 |
| 3 | 84 ÷ 22 = 3 remainder 18 |
| 4 | 22 ÷ 18 = 1 remainder 4 |
| 5 | 18 ÷ 4 = 4 remainder 2 |
| 6 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 187 and 16 | 1 |
| 104 and 168 | 8 |
| 197 and 156 | 1 |
| 88 and 84 | 4 |
| 17 and 167 | 1 |