Greatest Common Divisor (GCD) of 84 and 144
The greatest common divisor (GCD) of 84 and 144 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 144?
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 ÷ 144 = 0 remainder 84 |
| 2 | 144 ÷ 84 = 1 remainder 60 |
| 3 | 84 ÷ 60 = 1 remainder 24 |
| 4 | 60 ÷ 24 = 2 remainder 12 |
| 5 | 24 ÷ 12 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 67 and 71 | 1 |
| 133 and 142 | 1 |
| 17 and 103 | 1 |
| 25 and 49 | 1 |
| 152 and 128 | 8 |