Greatest Common Divisor (GCD) of 117 and 185
The greatest common divisor (GCD) of 117 and 185 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 117 and 185?
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 | 117 ÷ 185 = 0 remainder 117 |
| 2 | 185 ÷ 117 = 1 remainder 68 |
| 3 | 117 ÷ 68 = 1 remainder 49 |
| 4 | 68 ÷ 49 = 1 remainder 19 |
| 5 | 49 ÷ 19 = 2 remainder 11 |
| 6 | 19 ÷ 11 = 1 remainder 8 |
| 7 | 11 ÷ 8 = 1 remainder 3 |
| 8 | 8 ÷ 3 = 2 remainder 2 |
| 9 | 3 ÷ 2 = 1 remainder 1 |
| 10 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 107 and 177 | 1 |
| 57 and 40 | 1 |
| 103 and 134 | 1 |
| 156 and 92 | 4 |
| 16 and 88 | 8 |