perplexus.info :: Numbers : Unit Fractions

Unit Fractions (Posted on 2006-10-13)

Call a fraction a "unit fraction" if it can be written as 1/n, where n is a positive integer.

How many more ways can the unit fraction 1/n be written as a sum of two (possibly equivalent) unit fractions than as a difference of two unit fractions?

Submitted by Gamer

Rating: 4.3333 (3 votes)

Solution: (Hide)

First, note that the equation (1/n) = (1/(n+a))+(1/(n/a)(n+a)) holds true because it equals (n+a)/n(n+a) = n/n(n+a) + a/n(n+a)

It also is true that 1/n = 1/(n-a) - 1/((n/a)(n-a)) because it equals (n-a)/(n(n-a)) = n/(n(n-a)) - a/(n(n-a))

Then, the restrictions in both cases are that a is an integer and n(n-a)/a is an integer, which simplifies to requiring a to be a factor of n² since n(n-a)/a equals (n²/a)-n.

The only difference is that a can equal n in the sum case (resulting in two equivalent fractions), but if a equalled n in the second case, it would result in division by zero.

So, the answer in all cases is 1.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-03-21 02:19:58
re: greater detail	brianjn	2006-10-17 19:37:31
greater detail	Dennis	2006-10-17 13:20:23
re: a solution	brianjn	2006-10-16 02:17:40
re: a solution	Gamer	2006-10-15 00:59:30
a solution	Dennis	2006-10-13 11:47:11

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