Greatest Common Divisor (GCD) of 110 and 77
The greatest common divisor (GCD) of 110 and 77 is 11.
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 110 and 77?
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 | 110 ÷ 77 = 1 remainder 33 |
| 2 | 77 ÷ 33 = 2 remainder 11 |
| 3 | 33 ÷ 11 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 30 and 41 | 1 |
| 49 and 38 | 1 |
| 133 and 117 | 1 |
| 116 and 178 | 2 |
| 108 and 47 | 1 |