Greatest Common Divisor (GCD) of 89 and 144
The greatest common divisor (GCD) of 89 and 144 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 89 and 144?
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 | 89 ÷ 144 = 0 remainder 89 |
| 2 | 144 ÷ 89 = 1 remainder 55 |
| 3 | 89 ÷ 55 = 1 remainder 34 |
| 4 | 55 ÷ 34 = 1 remainder 21 |
| 5 | 34 ÷ 21 = 1 remainder 13 |
| 6 | 21 ÷ 13 = 1 remainder 8 |
| 7 | 13 ÷ 8 = 1 remainder 5 |
| 8 | 8 ÷ 5 = 1 remainder 3 |
| 9 | 5 ÷ 3 = 1 remainder 2 |
| 10 | 3 ÷ 2 = 1 remainder 1 |
| 11 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 141 and 86 | 1 |
| 169 and 99 | 1 |
| 170 and 13 | 1 |
| 94 and 145 | 1 |
| 148 and 104 | 4 |