How many pairs of positive integers (a,b) with a+b≤100 satisfy the equation a+b^-1a^-1+b=13?

1992 AMC 12 Problem 13

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13.

How many pairs of positive integers

(a,b)

with

a+b\le100

satisfy the equation

\frac{a+b^{-1}}{a^{-1}+b}=13?

1

5

7

9

13

Multiply the numerator and denominator by

ab

After simplifying, write

a

as a multiple of

b

and apply the bound on

a+b

Multiplying the numerator and denominator by

ab

gives

\frac{a^2b+a}{b+ab^2} =\frac{a(ab+1)}{b(ab+1)} =\frac ab.

Thus

a=13b.

The condition

a+b\le100

becomes

14b\le100,

b

may be any integer from

1

through

7.

There are

7

pairs.

Thus the correct answer is C.