Should we consider multiplicity while solving this problem? I am trying to solve the problem :


A single fence is to be constructed from posts 6 inches wide and separated by lengths of chain 5 feet . If a certain fence begins and ends with a post.Which of following could be length of fence in feet ? a)17 b)28 c)35 d)39 e)50.  (Ans:a,b,d and e)


Since the total space from one fence to another would be 6ft. I believe the fence size should be a multiple of 6 .None of the nos here are multiple of 6. How did they get the answer above ?
 A: Let $n$ be the number of posts.  Then there are $n-1$ lengths of chain, since there has to be a post at both ends. Because the post is $\frac{1}{2}$ of a foot, the total length of the fence, in feet, is 
$$\frac{1}{2}n +5(n-1).$$
This is $5.5n -5$.  So we want to check which of our numbers is of the shape $5.5n-5$, for some integer $n$.
For example, can we have $5.5n-5=17$? Sure, $22$ is an integer multiple of $5.5$.
In general, we are interested in whether, for given $k$,  there is an integer $n$ such that $5.5n-5=k$, or equivalently $11n=2k+10$. So what we care about is whether $11$ divides $2k+10$.
A: The inital length is 6ft, but every time you add a post after that it increases by 5.5ft
|=post
--=chain
|--| is 6ft
but
|--|--| is 11.5ft, which isn't divisible by 6 obviously. Adding another post, makes it 17ft.
|--|--|--|
Knowing this, you can make a model. Keep in mind the number of posts will always be 1 more than the number of chains.
x=#posts
x(.5) + (x-1)(5) = length
.5 is length of post, 5 is length of chain.
Subbing in for the available lengths, and solving for x, will give you the number of posts each length requires, if it is a whole number it works, if x comes out as a fraction, then it is not a valid fence.
