Does adding an integer to a simplified fraction ensure it will always be simplified

Question

$${A + BC\over B}$$ Assuming C is an integer and A divided by B is a simplified equation. Will this equation always be simplified? and is there a mathematical proof?

Answer 1

Short answer:

Yes, the fraction will alsways stay simplified, since $\gcd(A,B)=\gcd(A+BC,B)=1$.

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Longer answer:

• $\frac{A}{B}$ is simplified, so there is no prime $p$ dividing both $A,B$.
• We will show that there is no prime $p$ dividing both $A+BC$ and $B$, this will tell us that the fraction $\frac{A+BC}{B}$ is simplified aswell.

Suppose, for sake of contradiction, that there is a prime number $p$ dividing both $A+BC$ and $B$. Then $p$ will also divide $(A+BC)+(B)$, since both the numbers in the parentheses are divisible by $p$. In general, $p$ will also divide $$(A+BC)+(B)(-C)=A$$ by the same argument. This means that $p$ divides both $A$ and $B$ which is impossible by assumption. We must therefore have that $p$ is a unit (not a prime), that is, $p=\pm1$.