impossible answer: nobody will get (math)?

Question

Suppose that p/q is a rational number such that
137/2008 < p/q < 137/2007, and that p and q
are relatively prime positive integers.
a. What is the least possible value for q?
b. What is the 2007th smallest possible value for q?

I got as far as

0.06822...< x < 0.06826
like the title says this is like impossible to get!

Answer 1

p/q = 137 / (2007 + 1/n), n ≥ 2
= 137*n / (2007n + 1)
Note that since 137 is prime, 137n can only be an integer if n is an integer.

p = 137*n and q = 2007*n + 1
p has factors 137 and the factors of n
q is not divisible by n or any of n's factors.
BUT q may be divisible by 137 for certain values of n.
We need to find these values.

Using a brute force approach I found that there are 15 values of n (< 2009) for which 2007*n + 1 is divisible by 137 (or for which 89*n + 1 is divisible by 137).

(These values of n are 20, 157, 294, 431, 568, 705, 842, 979, 1116, 1253, 1390, 1527, 1664, 1801, 1938). I'm sure someone can find a better mathematical approach to find this.

The smallest value of q will occur when n is one of these values because p and q will be reduce to:
p = n, q = (2007*n + 1)/137

So the smallest value of q will be when n=20, ie
q_min = 293, (corresponding p = 20).
*******************************************

This now makes finding the 2007th value very difficult since we need a longer list of n values for which 2007*n + 1 is divisible by 137.
For example, for n=705, we get p/q = 705 / 10328
But for n = 2, we get p/q = 274 / 4015
So we need to combine these until we find the 2007th value.

Here's another complication. For n = 705, our formula gives
p/q = 705 / 10328
It turns out that 705 / 10329 also satisfies the condition.
*******************************************

OK I wrote a very simple program (< 10 lines of script) to work this out.
The 2007th smallest value of q is 10885 (p = 743).
This program also verifies that q = 293 (p = 20) is indeed the smallest possible value of q.
Do you also want the 2008th smallest value? It's 10886.

Answer 2

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Answer 3

23/337 = .06825 There might be a smaller q since I was under the assumption p and q were prime.
Okay, I have to ask this question before I spend sleepless nights on this problem. Are you sure this isn't a computer program assignment?
If I don't get this, would you send me the solution when you get it please?

Answer 4

me neither imagine having to solve a bunch of letters that is all we have lie ebasdberjfverfvuyrfvuer and that is the problem