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Perfect Square To Divisibility By 56 (Posted on 2010-04-01) Difficulty: 4 of 5
N is a positive integer such that each of 3*N + 1 and 4*N + 1 is a perfect square.

Is N always divisible by 56?

If so, prove it. Otherwise, give a counterexample.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Some Thoughts Does this explain why 7 is a factor of n? | Comment 12 of 14 |

 

1.       56 = 7*8. The part of the solution showing 8 is a factor has already been done.<o:p></o:p>

2.       For all m, m^2-1 = (m-1)(m+1)<o:p></o:p>

3.       3n+1 is a perfect square, x ^2. 3n= (x-1)(x+1) = k<o:p></o:p>

4.       4n+1 is a perfect square, y^2.  4n= (y-1)(y+1) = l<o:p></o:p>

5.       7n = k+l<o:p></o:p>

6.       n=(k+l)/7<o:p></o:p>

7.       n is an integer. Therefore n is divisible by 7.<o:p></o:p>

 


  Posted by broll on 2010-04-04 04:09:42
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