Greatest Common Divisor (GCD) of 172 and 107
The greatest common divisor (GCD) of 172 and 107 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 107?
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 ÷ 107 = 1 remainder 65 |
| 2 | 107 ÷ 65 = 1 remainder 42 |
| 3 | 65 ÷ 42 = 1 remainder 23 |
| 4 | 42 ÷ 23 = 1 remainder 19 |
| 5 | 23 ÷ 19 = 1 remainder 4 |
| 6 | 19 ÷ 4 = 4 remainder 3 |
| 7 | 4 ÷ 3 = 1 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 23 and 62 | 1 |
| 173 and 135 | 1 |
| 35 and 105 | 35 |
| 129 and 104 | 1 |
| 126 and 36 | 18 |