# Quantitative Aptitude – Algebra – Number of integer solutions – 1/a + 1/b = 1/9

## Question

How many different pairs (a, b) of positive integers are there such that a ≤ b and
1/a + 1/b = 1/9

3

## Solution

From CAT 2017 – Quantitative Aptitude – Algebra – Number of integer solutions, we can see that,
9(a + b) = ab
ab – 9a – 9b + 81 = 81
(a – 9) (b – 9) = 81 = 34
As a, b > 0 and a ≤ b, there are only 3 ordered pairs, given by a – 9 = 1, 3 or 9 and correspondingly b – 9 = 81, 27, 9.
We have to make sure that we satisfy the condition, a≤b

These are the following pairs of (a,b) that satisfy the condition
(a,b) = (10,90), (12, 36), (18,18)

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## One Reply to “Quantitative Aptitude – Algebra – Number of integer solutions – 1/a + 1/b = 1/9”

1. Rishikaa says:

1/a+1/b=9
a+b/ab=9
9ab-a-b+81=81
9(ab-9)-(a+b)=81