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\begin{align*} 5x + 10 = \frac{x}{3} + 24 \\ (A) 1 \\ (B) 2 \\ (C) 3 \\ (D) 4 \\ (E) 5 \\ \end{align*} The problem implies that I can deduce the answer $C, 3$ without solving the equation or testing the five choices. I can't see how.

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2 Answers 2

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Given that all the possibilities are integers, the solution must be divisible by $3$

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  • $\begingroup$ Feels so obvious now. Learning the rules and application of methods in maths isn't proving so difficult. But the ability to apply logic and reason improves a little slower. I hope to get a lot better by the time I meet calculus. I'm doing problems until my fingers bleed. $\endgroup$
    – user839943
    Mar 21, 2021 at 20:04
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    $\begingroup$ @Edmund in all fairness, it probably took me longer to think of that than to check for the correct one $\endgroup$ Mar 21, 2021 at 20:08
  • $\begingroup$ @Edmund You should check out the Art of Problem Solving books, they contain enough problems to make your fingers bleed $\endgroup$
    – Some Guy
    Mar 21, 2021 at 21:34
  • $\begingroup$ @ me if you find them helpful $\endgroup$
    – Some Guy
    Mar 21, 2021 at 21:35
  • $\begingroup$ @SomeGuy AOPS is my current resource. I'm on the second book now "Introduction to Algebra". I used Khan academy to complete pre-k-8th grade. $\endgroup$
    – user839943
    Mar 21, 2021 at 22:05
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The left side is always an integer because the input answers are also always integers. Thus, the right side must be an integer, and for $\frac x3$ to be an integer, $x$ must be divisible by $3$ as marla stated.

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