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Topic: Lines through a point (Read 330 times) |
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Hooie
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Lines through a point
« on: Mar 4th, 2004, 10:44pm » |
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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)?
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Sir Col
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impudens simia et macrologus profundus fabulae
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Re: Lines through a point
« Reply #1 on: Mar 5th, 2004, 12:48am » |
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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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Sir Col
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Re: Lines through a point
« Reply #2 on: Mar 5th, 2004, 12:51am » |
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I saw that, T&B! Which means you probably saw mine too. It'll be our little secret! 
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ThudnBlunder
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Re: Lines through a point
« Reply #3 on: Mar 5th, 2004, 12:53am » |
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Quote:| I saw that, T&B! Which means you probably saw mine too. |
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I sure did! 
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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kellys
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Re: Lines through a point
« Reply #4 on: Mar 5th, 2004, 12:43pm » |
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Is it wrong to add redundant proofs? I just need the practice... Here's a # theoretic proof:
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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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| « Last Edit: Mar 5th, 2004, 12:46pm by kellys » |
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