<s>[INST] Finding the largest possible value for the number of elements If $S$ is a subset  from the set {$1,2,3,...,20$} such that the sum of any two elements is not divisible by $5$.How to find the largest possible value for the number of elements for $S$
 [/INST] If we choose at least one number of the form $5a+1,$ we can not choose any number of the form $5b+4$ and vice versa where $a,b$ are non-negative integers. 
If we choose at least one number of the form $5c+2$ we can not choose any number of the form $5d+3$  and vice versa where $c,d$ are non-negative integers. 
Clearly, we can choose at most one number of the form $5e$ as $5\mid(5e_1+5e_2)$
Observe that $5\not\mid(5a_1+1+5a_2+1),5\not\mid(5a_1+1+5b_1+2),5\not\mid(5b_1+2+5b_2+2)$ where $a_i,b_i$ are non-negative integers.
As $0<5a+1\le 20\implies 0\le a\le3,$
$0<5b+4\le20\implies 0\le b\le3$ 
and $0<5e\le 20\implies 1\le e\le 4,$
we can at most choose $4+4+1=9$ elements
</s>