Greatest Common Divisor (GCD) of 105 and 48
The greatest common divisor (GCD) of 105 and 48 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 105 and 48?
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 | 105 ÷ 48 = 2 remainder 9 |
| 2 | 48 ÷ 9 = 5 remainder 3 |
| 3 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 50 and 16 | 2 |
| 97 and 48 | 1 |
| 107 and 122 | 1 |
| 192 and 69 | 3 |
| 153 and 200 | 1 |