Greatest Common Divisor (GCD) of 115 and 188
The greatest common divisor (GCD) of 115 and 188 is 1.
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 115 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 | 115 ÷ 188 = 0 remainder 115 |
| 2 | 188 ÷ 115 = 1 remainder 73 |
| 3 | 115 ÷ 73 = 1 remainder 42 |
| 4 | 73 ÷ 42 = 1 remainder 31 |
| 5 | 42 ÷ 31 = 1 remainder 11 |
| 6 | 31 ÷ 11 = 2 remainder 9 |
| 7 | 11 ÷ 9 = 1 remainder 2 |
| 8 | 9 ÷ 2 = 4 remainder 1 |
| 9 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 103 and 175 | 1 |
| 189 and 98 | 7 |
| 44 and 75 | 1 |
| 196 and 57 | 1 |
| 148 and 97 | 1 |