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Multiplication Table (Posted on 2006-07-07) Difficulty: 3 of 5
Imagine a multiplication table (like the one below, except it continues on forever):

   1   2   3   4     
 +---+---+---+---+---
1| 1 | 2 | 3 | 4 |...
 +---+---+---+---+---
2| 2 | 4 | 6 | 8 |...
 +---+---+---+---+---
3| 3 | 6 | 9 | 12|...
 +---+---+---+---+---
4| 4 | 8 | 12| 16|...
 +---+---+---+---+---
 |...|...|...|...|...

Find three of the same number in a straight line somewhere within the table. If this is not possible, show why not.

No Solution Yet Submitted by tomarken    
Rating: 3.0000 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
another solution | Comment 6 of 12 |
my interpretation of the problem is that for the multiplication of x and y the point (x,y) is used.  thus we are looking for 3 points
(a,n/a)    (b,n/b)   (c,n/c)
such that a,b and c are all unique divisors of a certain positive integer n.
now the equation from the first point to the second point is given by
y=(n/ab)x-(n/b)+(n/a)

so for the 3 to be colinear the 3rd one must be on this line thus

n/c=(n/ab)c-(n/b)+(n/a)
1/c=(c/ab)-(1/b)+(1/a)
1=(c^2/ab)-(c/b)+(c/a)
ab=c^2-ac+bc
ab+ac=c^2+bc
a(b+c)=c(c+b)
a=c
but a can not equal c thus according to this intepretation of the problem it is impossible


  Posted by Daniel on 2006-07-07 16:32:07
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