Greatest Common Divisor (GCD) of 75 and 48
The greatest common divisor (GCD) of 75 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 75 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 | 75 ÷ 48 = 1 remainder 27 |
| 2 | 48 ÷ 27 = 1 remainder 21 |
| 3 | 27 ÷ 21 = 1 remainder 6 |
| 4 | 21 ÷ 6 = 3 remainder 3 |
| 5 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 141 and 132 | 3 |
| 148 and 158 | 2 |
| 107 and 172 | 1 |
| 172 and 89 | 1 |
| 163 and 59 | 1 |