Greatest Common Divisor (GCD) of 45 and 110
The greatest common divisor (GCD) of 45 and 110 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 110?
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 ÷ 110 = 0 remainder 45 |
| 2 | 110 ÷ 45 = 2 remainder 20 |
| 3 | 45 ÷ 20 = 2 remainder 5 |
| 4 | 20 ÷ 5 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 189 and 193 | 1 |
| 181 and 12 | 1 |
| 112 and 92 | 4 |
| 136 and 191 | 1 |
| 164 and 144 | 4 |