Greatest Common Divisor (GCD) of 54 and 105
The greatest common divisor (GCD) of 54 and 105 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 54 and 105?
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 ÷ 105 = 0 remainder 54 |
| 2 | 105 ÷ 54 = 1 remainder 51 |
| 3 | 54 ÷ 51 = 1 remainder 3 |
| 4 | 51 ÷ 3 = 17 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 77 and 106 | 1 |
| 190 and 64 | 2 |
| 120 and 31 | 1 |
| 135 and 191 | 1 |
| 29 and 192 | 1 |