How many pairs of positive integers e, f satisfy 1/e + 4/f= 1/12 wher…

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:

$f$	$e = \frac{12f}{f-48}$​	Integer?
49	$588/1 = 588$	Yes
51	$612/3 = 204$	Yes
53	$636/5 = 127.2$	No
55	$660/7 \approx 94.3$	No
57	$684/9 = 76$	Yes
59	$708/11 \approx 64.4$	No

Step 3: Count valid pairs

Valid values of f:

$49,\ 51,\ 57$

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