Greatest Common Divisor (GCD) of 178 and 196
The greatest common divisor (GCD) of 178 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 178 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 | 178 ÷ 196 = 0 remainder 178 |
| 2 | 196 ÷ 178 = 1 remainder 18 |
| 3 | 178 ÷ 18 = 9 remainder 16 |
| 4 | 18 ÷ 16 = 1 remainder 2 |
| 5 | 16 ÷ 2 = 8 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 110 and 105 | 5 |
| 61 and 61 | 61 |
| 12 and 165 | 3 |
| 135 and 146 | 1 |
| 45 and 131 | 1 |