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Given $\frac{a}{A}$ and $\frac{b}{B}$ be two simplified fractions and $\operatorname{lcd}(A,B)$ be the Least Common Denominator of $A$ and $B$. Let $\frac{a}{A}+\frac{b}{B}=\frac{c}{\operatorname{lcd}(A,B)}$

Question:

in what scenario $\frac{c}{\operatorname{lcd}(A,B)}$ may be further simplified, or put another way, there may exist a common factor between $c$ and $\operatorname{lcd}(A,B)$

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    Hint: $1/3 + 1/6 = 1/2$. – 2017-01-02
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    I think you mean "lcm" rather than "lcd"; "lcd" is a liquid-crystal diode; "lcm" is the "least common multiple". : ) – 2017-01-02
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    LCD can mean Least Common Denominator – 2017-01-02
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    @dxiv: thanks for the example. can there be a proof? – 2017-01-02
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    @techie11 Sorry, proof of what? It's not clear what is the statement that you mean to prove. The title asks `can sum ... be further simplified?` and the example I gave proves that the answer is `yes`. – 2017-01-02
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    @dxiv 5: I rephrased the problem. hope it clarifies. thanks – 2017-01-03
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    @techie11 The sum fraction simplifies iff $\gcd(aB+bA,AB) \gt \gcd(A,B)$ and I don't know that you'll find a much simpler condition. – 2017-01-03
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    @dxiv 5: Thanks. – 2017-01-04

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Counter example:

$$\frac{1}{4}+\frac{1}{6}=\frac{5}{12}$$

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    I think this is true in most cases. it is rare that such fraction can be further simplified. like the example @dxiv 5 gave. – 2017-01-03