Greatest Common Divisor (GCD) of 178 and 106
The greatest common divisor (GCD) of 178 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 178 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 | 178 ÷ 106 = 1 remainder 72 |
| 2 | 106 ÷ 72 = 1 remainder 34 |
| 3 | 72 ÷ 34 = 2 remainder 4 |
| 4 | 34 ÷ 4 = 8 remainder 2 |
| 5 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 67 and 92 | 1 |
| 108 and 94 | 2 |
| 122 and 136 | 2 |
| 170 and 55 | 5 |
| 41 and 170 | 1 |