
Greatest Common Divisor (GCD) of 119 and 45
The greatest common divisor (GCD) of 119 and 45 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 119 and 45?
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 | 119 ÷ 45 = 2 remainder 29 |
2 | 45 ÷ 29 = 1 remainder 16 |
3 | 29 ÷ 16 = 1 remainder 13 |
4 | 16 ÷ 13 = 1 remainder 3 |
5 | 13 ÷ 3 = 4 remainder 1 |
6 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
Numbers | GCD |
---|---|
167 and 77 | 1 |
25 and 191 | 1 |
192 and 188 | 4 |
86 and 44 | 2 |
108 and 162 | 54 |