**Quantitative Aptitude – Algebra – Number of integer solutions**

**Question**

How many different pairs (a, b) of positive integers are there such that a ≤ b and

1/a + 1/b = 1/9

**Answer**

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)

Answer: 3

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1/a+1/b=9

a+b/ab=9

9ab-a-b+81=81

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