Answer to Question #147262 in Combinatorics | Number Theory for Akrix Salram

Question

Let us consider two irreducible fractions. The denominator of the first one is equal to 8200,and the denominator of the second to 4300. What is the smallest possible denominator of a fraction equal to the sum of these fractions, after the fraction is reduced? (For example, (2/3) + (8/15) = (18/15) = (6/5), and the denominator after the reduction is equal to 5.)

Answer 1

We have

$\frac{x}{8200}+\frac{y}{4300}=\frac{43x+82y}{352600} \\ \text{But the gcd(43x+82y,352600)} = \frac{(43x+82y)352600}{lcm(43x+82y,352600)}, \text{ then for every } x,y\in \mathbb{Z} \text{ gcd(43x+82y,352600) divides (43x+82y) and 352600} \\ \therefore \boxed{\text{ the lowest denominator is } \frac{352600}{gcd(43x+82y,352600)}=\frac{lcm(43x+82y,352600)}{43x+82y} } \\ \text{For example if x=y=1 then we will have the lowest denominator to be } \frac{lcm(43+82,352600)}{43+82}= \frac{lcm(125,352600)}{125}=\frac{1763000}{125}=14104$