Greatest Common Divisor (GCD) of 117 and 192
The greatest common divisor (GCD) of 117 and 192 is 3.
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 192?
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 ÷ 192 = 0 remainder 117 |
| 2 | 192 ÷ 117 = 1 remainder 75 |
| 3 | 117 ÷ 75 = 1 remainder 42 |
| 4 | 75 ÷ 42 = 1 remainder 33 |
| 5 | 42 ÷ 33 = 1 remainder 9 |
| 6 | 33 ÷ 9 = 3 remainder 6 |
| 7 | 9 ÷ 6 = 1 remainder 3 |
| 8 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 199 and 171 | 1 |
| 134 and 161 | 1 |
| 43 and 37 | 1 |
| 135 and 109 | 1 |
| 153 and 57 | 3 |