Subject: Re: What the heck?
Date: Thu, 3 Jul 1997 20:31:36 -0400
From: Alexander Bogomolny

Dear Anthony:

I spent very happy 5 years in Iowa City. I fondly remember a small university town with exceptionally cultured population that usually looked up to the faculty of the university. Upon moving to New Jersey, it was a shock for me to discover how different America may be.

Are you originally from the East Coast by any chance?

The problem you have asked me to solve is

1/x + 1/y = 1/7 How?
what the heck?

The problem is ill posed in several ways. I hope you get my meaning. But returning to the math, an equation with two unknowns has infintely many solutions: (x,y) = (1, -7/6), (x,y) = (1/2, -7/13), etc.

With a lot of good will I may surmise that you are looking for integer solutions. Note that it does not follow from your formulation though. Unless, of course, "heck" is the modern midwestern slang for integer. In which case I should wonder why haven't you asked "What the hecks?"

In integers, this equation has a single solution: x = y = 14.

The proof is as follows. For one, both x and y are greater than 7. So let x = 7+a and y = 7+b. Simplifying the equation we get

7(7+b) + 7(7+a) = (7+a)(7+b).

Further simplification results in 49 = ab. It's easy to see we can't have a = 49 and b = 1. So a = b = 7 is the only possibility.

Regards,
Alexander Bogomolny

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