Egyptian fractions - Math Central

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Question from vivek, a student:

An Egyptian fraction has numerator equal to 1, and its denominator is a positive integer. What is the maximum number of different egyptian fractions
such that their sum is equal to 1, and their denominator are equal to 10 or less?

Hi Vivek,

Some of the combinations are obvious:
1/2+1/2=1
1/3+1/3+1/3=1
1/4+1/4+1/4+1/4=1 and so on.

Remember that
1/2 = 1/4+1/4=1/6+1/6+1/6= 1/8+1/8+1/8+1/8=1/10+1/10+1/10+1/10+1/10

So we have other combinations like
1/2+1/4+1/4=1
1/2+1/6+1/6+1/6=1
1/2+1/8+1/8+1/8+1/8=1
1/2+1/10+1/10+1/10+1/10+1/10=1
So there are 5 combinations where one of the terms is 1/2 but we also could have figured this out since there are 5 equivalent expressions to 1/2.

You can follow the same method for the other Egyptian fractions. [Hint: Equivalent fractions will have at least one common factor]
Janice

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