Greatest Common Divisor (GCD) of 84 and 132
The greatest common divisor (GCD) of 84 and 132 is 12.
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 84 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 | 84 ÷ 132 = 0 remainder 84 |
| 2 | 132 ÷ 84 = 1 remainder 48 |
| 3 | 84 ÷ 48 = 1 remainder 36 |
| 4 | 48 ÷ 36 = 1 remainder 12 |
| 5 | 36 ÷ 12 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 104 and 177 | 1 |
| 170 and 174 | 2 |
| 168 and 63 | 21 |
| 120 and 105 | 15 |
| 128 and 79 | 1 |