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Download Problem #518 Solution Find all pairs of positive integers x, y such
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Transcript
Problem #518
Solution
Find all pairs of positive integers x, y such that
1 1
1
+ = .
x y
10
Answer. The solutions are the ordered pairs (x, y) in the
{(11, 110), (12, 60), (14, 35), (15, 30), (20, 20), (40, 15), (35, 14), (60, 12), (110, 11)}.
Solution. The given equation is equivalent to
(x − 10)(y − 10) = 100.
This means that x − 10 is a divisor of 100 and y − 10 is 100/(x − 10). The
divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. This leads to the ordered
pairs in the answer.
Note that for general N , the number of solutions of
1 1
1
+ =
x y
N
is d(N 2 ); i.e., the number of divisors of N 2 .
Source: Underwood Dudley, Elementary Number Theory.
