Greatest Common Divisor (GCD) of 77 and 128
The greatest common divisor (GCD) of 77 and 128 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 77 and 128?
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 | 77 ÷ 128 = 0 remainder 77 |
| 2 | 128 ÷ 77 = 1 remainder 51 |
| 3 | 77 ÷ 51 = 1 remainder 26 |
| 4 | 51 ÷ 26 = 1 remainder 25 |
| 5 | 26 ÷ 25 = 1 remainder 1 |
| 6 | 25 ÷ 1 = 25 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 56 and 130 | 2 |
| 184 and 134 | 2 |
| 80 and 126 | 2 |
| 110 and 84 | 2 |
| 46 and 59 | 1 |