Question #147262
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.)
1
Expert's answer
2020-12-02T12:02:18-0500

We have

x8200+y4300=43x+82y352600But the gcd(43x+82y,352600)=(43x+82y)352600lcm(43x+82y,352600), then for every x,yZ gcd(43x+82y,352600) divides (43x+82y) and 352600 the lowest denominator is 352600gcd(43x+82y,352600)=lcm(43x+82y,352600)43x+82yFor example if x=y=1 then we will have the lowest denominator to be lcm(43+82,352600)43+82=lcm(125,352600)125=1763000125=14104\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


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