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A Deja Vu Problem All Over Again


How many pairs of positive fractions with denominators 3 and 5 have a sum of 2 1/15?

Note: Assume the numerators are integers.

New Task: Repeat the problem, but remove the restriction "positive."

New New Task: How many pairs of positive fractions with denominators A and B have a sum of N/(AB), for A,B, N nonzero? (Also, can you find A and B such that either no pairs or a finite number of pairs exist?)

 

Source: Adapted from G. Merriman's To Discover Mathematics, 1942, p. 367


Hint: You can try some possible numerators....but, how can you know if you have them all?

How can you attack the problem in general terms...think, algebra to the rescue?

 


Solution Commentary: You can play with this on your own....but ideas such as a linear equation aA+bB = n should eventually appear.

Now, where have you seen this before...and how was that "brief meeting" helpful?