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Subject: "Number theory question"     Previous Topic | Next Topic
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shivgaur
Member since Mar-9-11
Mar-20-11, 11:29 AM (EST)
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"Number theory question"
 
   The Problem:
Suppose for a positive integer n both 5n+1 and 7n+1 are perfect squares. Show that n is divisible by 24

My attempt:
Since 5n+1 is a perfect square and n is a positive integer then, the n's for which 5n+1 is a perfect square are: n= 3, 7, 16, 24 ..........
and for 7n+1 : n = 5, 9, 24......
therefore the least common 'n' which makes both 5n+1 and 7n+1 a perfect square is 24 and therefore 'n' is divisible by 24.

Any better approach to this problem?

Shiv


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alexb
Charter Member
2792 posts
Mar-23-11, 06:46 PM (EST)
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1. "RE: Number theory question"
In response to message #0
 
   This only says that n = 24 works. It does not prove the general statement.


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shivgaur
Member since Mar-9-11
Mar-26-11, 08:35 PM (EST)
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2. "RE: Number theory question"
In response to message #1
 
   Thanks Alex! Got the point. Will try for a general solution.

Shiv


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mr_homm
Member since Jan-5-11
Mar-29-11, 07:20 AM (EST)
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3. "RE: Number theory question"
In response to message #2
 
   A hint:

Try reducing the expressions mod 3 and mod 8. Then, what are all the squares in mod 3 and mod 8? From there, you can work out the details.


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shivgaur
Member since Mar-9-11
Apr-05-11, 11:22 AM (EST)
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4. "RE: Number theory question"
In response to message #3
 
   Many thanks! mod 3 and mod 8 works out beautifully!

Shiv


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mr_homm
Member since Jan-5-11
Apr-07-11, 11:54 AM (EST)
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5. "RE: Number theory question"
In response to message #4
 
   You're welcome. I'm glad to hear that. It is sometimes difficult to give just the right amount of hint. Too little and there is frustration, too much and the fun is spoiled.


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