Answer:
(A, B) = (4, 12) or (12, 4)
Step-by-step explanation:
An Egyptian fraction is the sum of distinct unit fractions. Here, we're asked to decompose 1/3 into an Egyptian fraction sum of two fractions. There are formulas available when the sum has a numerator value of 2.
Here, the sum is a unit fraction. A reasonable approach is to use 1/(n+1) as the larger portion of 1/n. Then the other portion is 1/(n(n+1)). That is effectively what we end up with here.
__
Multiplying by 3AB gives ...
AB = 3B +3A
Solving for B, we find ...
AB -3B = 3A
B(A -3) = 3A
B = 3A/(A -3)
In order for A-3 to be a factor of A, or equal to 3, we must have ...
A-3 = 1 ⇒ A = 4
A-3 = 3 ⇒ A = 6
A-3 = 3×3 ⇒ A = 4×3
This will give integer values for B when A is one of 4, 6, or 12. In the case of A=6, the two fractions are equal, which is not what you want
The two solutions are ...
(A, B) = (4, 12) or (12, 4)
