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anemone
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Find all positive integers $a$ and $b$ such that $\dfrac{a^2+b}{b^2-a}$ and $\dfrac{b^2+a}{a^2-b}$ are both integers.
topsquark said:[sp]
Since you got the problem stuck into my head, this is my revenge!
IT'S A SMALL WORLD AFTER ALL IT'S A SMALL WORLD AFTER ALL IT'S A SMALL WORLD AFTER ALL IT'S A SMALL SMALL WORLD!
Now that that's stuck in your head I can finally relax.
[/sp]
-Dan
anemone said:Find all positive integers $a$ and $b$ such that $\dfrac{a^2+b}{b^2-a}$ and $\dfrac{b^2+a}{a^2-b}$ are both integers.
Integer equations are mathematical expressions that involve whole numbers, also known as integers. These equations typically contain variables, such as $a$ and $b$, and require solving for the unknown values.
To solve integer equations with $a$ and $b$, you can use basic algebraic principles such as combining like terms, using inverse operations, and applying the distributive property. It is important to follow the order of operations and always check your solution by substituting the values back into the original equation.
Linear integer equations involve variables that are raised to the first power, while quadratic integer equations involve variables that are raised to the second power. This means that quadratic equations can have two solutions, while linear equations have only one solution.
No, the method used to solve integer equations may vary depending on the type of equation. For example, linear equations can be solved using the substitution or elimination method, while quadratic equations can be solved using factoring or the quadratic formula.
Checking your solution is important because it ensures that you have found the correct values for the variables. Sometimes, a solution may seem correct but may not satisfy the original equation. Checking also helps to catch any mistakes made during the solving process.