Greatest Common Divisor (GCD) of 183 and 105
The greatest common divisor (GCD) of 183 and 105 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 183 and 105?
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 | 183 ÷ 105 = 1 remainder 78 |
| 2 | 105 ÷ 78 = 1 remainder 27 |
| 3 | 78 ÷ 27 = 2 remainder 24 |
| 4 | 27 ÷ 24 = 1 remainder 3 |
| 5 | 24 ÷ 3 = 8 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 200 and 158 | 2 |
| 11 and 94 | 1 |
| 42 and 40 | 2 |
| 159 and 45 | 3 |
| 91 and 199 | 1 |