Greatest Common Divisor (GCD) of 54 and 90
The greatest common divisor (GCD) of 54 and 90 is 18.
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 54 and 90?
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 | 54 ÷ 90 = 0 remainder 54 |
| 2 | 90 ÷ 54 = 1 remainder 36 |
| 3 | 54 ÷ 36 = 1 remainder 18 |
| 4 | 36 ÷ 18 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 200 and 181 | 1 |
| 57 and 75 | 3 |
| 123 and 101 | 1 |
| 47 and 37 | 1 |
| 34 and 147 | 1 |