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        riddles >> easy >> Lines through a point
        
(Message started by: Hooie on Mar 4^th, 2004, 10:44pm)

Title: Lines through a point
Post by Hooie on Mar 4^th, 2004, 10:44pm

Here's a math problem I thought was pretty fun:

How many lines are there that have a y intercept that's a positive integer, have an x intercept that's a positive prime, and pass through the point (4,3)?

Title: Re: Lines through a point
Post by Sir Col on Mar 5^th, 2004, 12:48am

Indeed, it is a very nice problem, Hooie!

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It's best to think of this as a line passing through (4,3) and (p,0), where p is prime. The only points the line cannot pass through on the x-axis (p), such that the extended line will cut the y-axis above the x-axis are (2,0) and (3,0). For all other values of p the line will have a negative gradient: m=-3/(p–4), so the y intercept will have a positive value.

The equation of the line will be: (y–3)=(-3/(p–4))(x–4).

When x=0 (on y-axis), the y intercept, c=12/(p–4)+3.

When p=5, c=11.
When p=7, c=7.
When p=11, c~=4.71.
When p=13, c~=3.92.

For all values of p>13, the line will never be parallel with the x-axis, so it will never reach the next lowest integer, 3. Hence there are exactly two lines.
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Title: Re: Lines through a point
Post by Sir Col on Mar 5^th, 2004, 12:51am

I saw that, T&B! Which means you probably saw mine too. It'll be our little secret! ;)

Title: Re: Lines through a point
Post by THUDandBLUNDER on Mar 5^th, 2004, 12:53am

Quote:

I saw that, T&B! Which means you probably saw mine too.

I sure did! ;D

Title: Re: Lines through a point
Post by kellys on Mar 5^th, 2004, 12:43pm

Is it wrong to add redundant proofs? ;D I just need the practice... Here's a # theoretic proof:
::[hide]
Restating the conditions, we have a line
x/X +y/Y = 1,
Where Y=n \in Z+ is the y-intercept, and X=p, prime is the x-intercept. The last condition is,
4/p + 3/n = 1
Solving for n,
n = 3p/(p-4)
For no p does p-4 divide p, so either p-4 divides 3, or p-4=1. This implies p=7 or 5, so n=7 or 15.
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