Greatest Common Divisor (GCD) of 113 and 192
The greatest common divisor (GCD) of 113 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 113 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 | 113 ÷ 192 = 0 remainder 113 |
| 2 | 192 ÷ 113 = 1 remainder 79 |
| 3 | 113 ÷ 79 = 1 remainder 34 |
| 4 | 79 ÷ 34 = 2 remainder 11 |
| 5 | 34 ÷ 11 = 3 remainder 1 |
| 6 | 11 ÷ 1 = 11 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 143 and 10 | 1 |
| 199 and 23 | 1 |
| 100 and 105 | 5 |
| 23 and 146 | 1 |
| 16 and 167 | 1 |