Greatest Common Divisor (GCD) of 45 and 80
The greatest common divisor (GCD) of 45 and 80 is 5.
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 45 and 80?
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 | 45 ÷ 80 = 0 remainder 45 |
| 2 | 80 ÷ 45 = 1 remainder 35 |
| 3 | 45 ÷ 35 = 1 remainder 10 |
| 4 | 35 ÷ 10 = 3 remainder 5 |
| 5 | 10 ÷ 5 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 75 and 65 | 5 |
| 68 and 11 | 1 |
| 35 and 140 | 35 |
| 193 and 103 | 1 |
| 86 and 53 | 1 |