Greatest Common Divisor (GCD) of 169 and 103
The greatest common divisor (GCD) of 169 and 103 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 169 and 103?
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 | 169 ÷ 103 = 1 remainder 66 |
| 2 | 103 ÷ 66 = 1 remainder 37 |
| 3 | 66 ÷ 37 = 1 remainder 29 |
| 4 | 37 ÷ 29 = 1 remainder 8 |
| 5 | 29 ÷ 8 = 3 remainder 5 |
| 6 | 8 ÷ 5 = 1 remainder 3 |
| 7 | 5 ÷ 3 = 1 remainder 2 |
| 8 | 3 ÷ 2 = 1 remainder 1 |
| 9 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 149 and 81 | 1 |
| 122 and 109 | 1 |
| 51 and 190 | 1 |
| 160 and 34 | 2 |
| 78 and 51 | 3 |