How many pairs of positive integers e, f satisfy 1/e + 4/f= 1/12 where f is an odd integer less than 60?

3

The correct answer is Option (3) → 3

We are given:

$\frac{1}{e} + \frac{4}{f} = \frac{1}{12}$​

where

• e, f are positive integers
• f is odd and less than 60

Step 1: Rearrange the equation

$\frac{1}{e} = \frac{1}{12} - \frac{4}{f} = \frac{f - 48}{12f}$

So,

$e = \frac{12f}{f - 48}$​

For e to be a positive integer, $f – 48$ must be a positive divisor of $12f$.

Thus,

$f > 48$

Step 2: Possible odd values of $f < 60$ and $>48$

$f = 49,\ 51,\ 53,\ 55,\ 57,\ 59$

Now check which give integer e:

Step 3: Count valid pairs

Valid values of f:

$49,\ 51,\ 57$

So there are 3 valid (e,f) pairs.