Greatest Common Divisor (GCD) of 110 and 188
The greatest common divisor (GCD) of 110 and 188 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 110 and 188?
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 | 110 ÷ 188 = 0 remainder 110 |
| 2 | 188 ÷ 110 = 1 remainder 78 |
| 3 | 110 ÷ 78 = 1 remainder 32 |
| 4 | 78 ÷ 32 = 2 remainder 14 |
| 5 | 32 ÷ 14 = 2 remainder 4 |
| 6 | 14 ÷ 4 = 3 remainder 2 |
| 7 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 191 and 138 | 1 |
| 197 and 54 | 1 |
| 200 and 199 | 1 |
| 29 and 19 | 1 |
| 180 and 159 | 3 |