Greatest Common Divisor (GCD) of 63 and 172
The greatest common divisor (GCD) of 63 and 172 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 63 and 172?
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 | 63 ÷ 172 = 0 remainder 63 |
| 2 | 172 ÷ 63 = 2 remainder 46 |
| 3 | 63 ÷ 46 = 1 remainder 17 |
| 4 | 46 ÷ 17 = 2 remainder 12 |
| 5 | 17 ÷ 12 = 1 remainder 5 |
| 6 | 12 ÷ 5 = 2 remainder 2 |
| 7 | 5 ÷ 2 = 2 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 81 and 55 | 1 |
| 107 and 115 | 1 |
| 188 and 155 | 1 |
| 132 and 180 | 12 |
| 160 and 116 | 4 |