Greatest Common Divisor (GCD) of 105 and 170
The greatest common divisor (GCD) of 105 and 170 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 105 and 170?
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 | 105 ÷ 170 = 0 remainder 105 |
| 2 | 170 ÷ 105 = 1 remainder 65 |
| 3 | 105 ÷ 65 = 1 remainder 40 |
| 4 | 65 ÷ 40 = 1 remainder 25 |
| 5 | 40 ÷ 25 = 1 remainder 15 |
| 6 | 25 ÷ 15 = 1 remainder 10 |
| 7 | 15 ÷ 10 = 1 remainder 5 |
| 8 | 10 ÷ 5 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 132 and 78 | 6 |
| 32 and 121 | 1 |
| 145 and 67 | 1 |
| 42 and 98 | 14 |
| 74 and 143 | 1 |