Greatest Common Divisor (GCD) of 78 and 112
The greatest common divisor (GCD) of 78 and 112 is 2.
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 78 and 112?
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 | 78 ÷ 112 = 0 remainder 78 |
| 2 | 112 ÷ 78 = 1 remainder 34 |
| 3 | 78 ÷ 34 = 2 remainder 10 |
| 4 | 34 ÷ 10 = 3 remainder 4 |
| 5 | 10 ÷ 4 = 2 remainder 2 |
| 6 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 197 and 20 | 1 |
| 116 and 19 | 1 |
| 157 and 60 | 1 |
| 161 and 97 | 1 |
| 186 and 67 | 1 |