Answer to Question #142442 in Algebra for Peace College

Question

Let us consider two irreducible fractions. The denominator of the first one is equal to 2200, and the denominator of the second to 10300. 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

Let the first fraction be $\dfrac{a}{2200}$ and the second fraction be $\dfrac{b}{10300}.$ Then

$$\dfrac{a}{2200}+\dfrac{b}{10300}=\dfrac{103a+22b}{22(103)(100)}$$

We have that $a$ is odd and $b$ i odd. Then the sum $103a+22b$ will be odd.

Since $a$ is not divisible by $22, 103a$ is not divisible by $22.$ Then the sum $103a+22b$ is not divisible by $22.$

Since $b$ is not divisible by $103, 22b$ is not divisible by $103.$ Then the sum $103a+22b$ is not divisible by $103.$

The greatest common factor of numerator and denominator may be $25.$ For example

$$\dfrac{1}{10300}+\dfrac{1}{2200}=\dfrac{22+103}{226600}=\dfrac{5}{9064}$$

The smallest possible denominator of a fraction is $9064.$