Greatest Common Divisor (GCD) of 34 and 51
The greatest common divisor (GCD) of 34 and 51 is 17.
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 34 and 51?
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 | 34 ÷ 51 = 0 remainder 34 |
| 2 | 51 ÷ 34 = 1 remainder 17 |
| 3 | 34 ÷ 17 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 178 and 125 | 1 |
| 83 and 200 | 1 |
| 100 and 48 | 4 |
| 200 and 115 | 5 |
| 106 and 20 | 2 |