Greatest Common Divisor (GCD) of 51 and 184
The greatest common divisor (GCD) of 51 and 184 is 1.
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 51 and 184?
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 | 51 ÷ 184 = 0 remainder 51 |
| 2 | 184 ÷ 51 = 3 remainder 31 |
| 3 | 51 ÷ 31 = 1 remainder 20 |
| 4 | 31 ÷ 20 = 1 remainder 11 |
| 5 | 20 ÷ 11 = 1 remainder 9 |
| 6 | 11 ÷ 9 = 1 remainder 2 |
| 7 | 9 ÷ 2 = 4 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 157 and 56 | 1 |
| 115 and 128 | 1 |
| 43 and 154 | 1 |
| 157 and 142 | 1 |
| 103 and 197 | 1 |