Greatest Common Divisor (GCD) of 172 and 105
The greatest common divisor (GCD) of 172 and 105 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 172 and 105?
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 | 172 ÷ 105 = 1 remainder 67 |
| 2 | 105 ÷ 67 = 1 remainder 38 |
| 3 | 67 ÷ 38 = 1 remainder 29 |
| 4 | 38 ÷ 29 = 1 remainder 9 |
| 5 | 29 ÷ 9 = 3 remainder 2 |
| 6 | 9 ÷ 2 = 4 remainder 1 |
| 7 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 140 and 84 | 28 |
| 190 and 155 | 5 |
| 12 and 135 | 3 |
| 156 and 157 | 1 |
| 160 and 200 | 40 |