Greatest Common Divisor (GCD) of 51 and 132
The greatest common divisor (GCD) of 51 and 132 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 51 and 132?
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 | 51 ÷ 132 = 0 remainder 51 |
| 2 | 132 ÷ 51 = 2 remainder 30 |
| 3 | 51 ÷ 30 = 1 remainder 21 |
| 4 | 30 ÷ 21 = 1 remainder 9 |
| 5 | 21 ÷ 9 = 2 remainder 3 |
| 6 | 9 ÷ 3 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 135 and 45 | 45 |
| 175 and 26 | 1 |
| 193 and 120 | 1 |
| 77 and 178 | 1 |
| 93 and 61 | 1 |