(x + a/b)(y – c/d) = x*y

 
 

(43 \; + \; 2/5) \,(31 \; - \; 2/7) \; = \; 43\times 31

(341 \; + \; 2/3) \,(205 \; - \; 2/5) \; = \; 341\times 205

(57 \; + \; 1/3) \,(43 \; - \; 1/4) \; = \; 57\times 43

(781 \; + \; 1/2) \,(521 \; - \; 1/3) \; = \; 781\times 521

 
 
Find an infinite set of products having the same property.

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

4 Responses to (x + a/b)(y – c/d) = x*y

  1. paul says:

    I found 427 such sets with 1<= a, b, c, d <=9.

    Each set can be produced with a sequence function for x and y and a corresponding one for (x * y).

    For example with a/b = 1/2 and c/d = 1/3 the sequence functions are
    3 n-2 , 2 n-1 , 6 n^2-7 n+2 for (x, y and x * y respectively).

    The 427 set sequence functions are here

    https://www.dropbox.com/s/ziuk1rncb3eiolc/x%20plus%20a%20over%20b.txt?dl=0

    Paul.

  2. paul says:

    The above example produces the following with n = 1 to 10.

    (1 + 1/2)(1 – 1/3) = 1 x 1
    (4 + 1/2)(3 – 1/3) = 4 x 3
    (7 + 1/2)(5 – 1/3) = 7 x 5
    (10 + 1/2)(7 – 1/3) = 10 x 7
    (13 + 1/2)(9 – 1/3) = 13 x 9
    (16 + 1/2)(11 – 1/3) = 16 x 11
    (19 + 1/2)(13 – 1/3) = 19 x 13
    (22 + 1/2)(15 – 1/3) = 22 x 15
    (25 + 1/2)(17 – 1/3) = 25 x 17
    (28 + 1/2)(19 – 1/3) = 28 x 19

    -2+3 n, -1+2 n, 2-7 n+6 n^2

    Paul.

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