Greatest Common Divisor (GCD) of 27 and 84
The greatest common divisor (GCD) of 27 and 84 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 27 and 84?
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 | 27 ÷ 84 = 0 remainder 27 |
| 2 | 84 ÷ 27 = 3 remainder 3 |
| 3 | 27 ÷ 3 = 9 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 80 and 37 | 1 |
| 176 and 85 | 1 |
| 160 and 63 | 1 |
| 179 and 150 | 1 |
| 184 and 60 | 4 |