# How is the answer to $5x + 10 = \frac{x}{3} + 24$ obvious without solving the equation or testing any of the choices (multi-choice answer)

\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.

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

• 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.
– user839943
Mar 21, 2021 at 20:04
• @Edmund in all fairness, it probably took me longer to think of that than to check for the correct one Mar 21, 2021 at 20:08
• @Edmund You should check out the Art of Problem Solving books, they contain enough problems to make your fingers bleed Mar 21, 2021 at 21:34
• @ me if you find them helpful Mar 21, 2021 at 21:35
• @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.
– user839943
Mar 21, 2021 at 22:05

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.