Greatest Common Divisor (GCD) of 40 and 78
The greatest common divisor (GCD) of 40 and 78 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 40 and 78?
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 | 40 ÷ 78 = 0 remainder 40 |
| 2 | 78 ÷ 40 = 1 remainder 38 |
| 3 | 40 ÷ 38 = 1 remainder 2 |
| 4 | 38 ÷ 2 = 19 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 167 and 67 | 1 |
| 193 and 52 | 1 |
| 103 and 139 | 1 |
| 75 and 137 | 1 |
| 180 and 173 | 1 |