Greatest Common Divisor (GCD) of 47 and 78
The greatest common divisor (GCD) of 47 and 78 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 47 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 | 47 ÷ 78 = 0 remainder 47 |
| 2 | 78 ÷ 47 = 1 remainder 31 |
| 3 | 47 ÷ 31 = 1 remainder 16 |
| 4 | 31 ÷ 16 = 1 remainder 15 |
| 5 | 16 ÷ 15 = 1 remainder 1 |
| 6 | 15 ÷ 1 = 15 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 153 and 183 | 3 |
| 11 and 43 | 1 |
| 35 and 117 | 1 |
| 74 and 180 | 2 |
| 136 and 108 | 4 |