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Is it possible to find the smallest positive integer/s that satisfy a given equation or some inequality?

Example: $2x^2-3x>24$

Is there a formula for this?

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closed as too broad by Aqua, Guy Fsone, mechanodroid, Arnaud D., kingW3 Oct 22 '17 at 14:41

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Start by checking if there is some small solution $a,b$. Then there are only finitely many possibilities to check for a smaller $a+b$. $\endgroup$ – hardmath Oct 18 '17 at 10:52
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So $5|a(a-1)$ and since $a$ and $a-1$ are relatively prime $5$ divides only one of them, say $a$. Since $a_{\min}=5$ you get $b = 2$ and so $(a+b)_{\min} = 7$.

If $5$ divides $a-1$ then $a_{\min} = 6$ which is not good.

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  • $\begingroup$ Thanks, but can you generalize the formula or prove that not all equations can be solved for the smallest integer? $\endgroup$ – Hyunsoo Kim Oct 18 '17 at 8:58
  • $\begingroup$ Then the answer is yes, if the equation has a solution in positive integers, since they are (positive integers) bounded below. $\endgroup$ – Aqua Oct 18 '17 at 9:01
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This is a very broad question. As for your specific equation, it's equivalent to $a(a-1) = 10b$ so you're basically looking for the smallest pair of successive numbers whose product is a multiple of 10. You also know that you need $a\geq5$ otherwise 5 can't divide $a(a-1)$, so you start looking at 5:

$5\times(5-1)=20=2\times10$. The smallest positive integers that satisfy your equation are $a=5$ and $b=2$.

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