Greatest Common Divisor (GCD) of 109 and 192
The greatest common divisor (GCD) of 109 and 192 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 109 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 | 109 ÷ 192 = 0 remainder 109 |
| 2 | 192 ÷ 109 = 1 remainder 83 |
| 3 | 109 ÷ 83 = 1 remainder 26 |
| 4 | 83 ÷ 26 = 3 remainder 5 |
| 5 | 26 ÷ 5 = 5 remainder 1 |
| 6 | 5 ÷ 1 = 5 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 86 and 183 | 1 |
| 62 and 118 | 2 |
| 160 and 66 | 2 |
| 59 and 113 | 1 |
| 143 and 170 | 1 |