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

Starting with the equation $\frac{1}{e} + \frac{4}{f} = \frac{1}{12}$

Isolate $\frac{1}{e}$:

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

Getting common denominator $(12f)$:

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

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

Therefore:

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

---

Rewriting $e$ by splitting the numerator:

$e = \frac{12(f - 48 + 48)}{f - 48}$

$e = \frac{12(f - 48)}{f - 48} + \frac{12(48)}{f - 48}$

$e = 12 + \frac{576}{f - 48}$

For $e$ to be a positive integer, $(f - 48)$ must be a divisor of $576$.

---

Finding the divisors of $576$:

$576 = 2^6 \times 3^2 = 64 \times 9$

The divisors of $576$ are: $1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576$

---

Applying the constraints:

• $f$ must be odd
• $f < 60$
• $f$ must be positive (which requires $f > 48$ for $e$ to be positive)

Since $f = 48 + \text{divisor}$, for $f$ to be odd, the divisor must be odd.

Odd divisors of $576$: $1, 3, 9$

---

Checking each odd divisor:

For divisor $= 1$: $f = 49$, $e = 12 + 576 = 588$ ✓

For divisor $= 3$: $f = 51$, $e = 12 + 192 = 204$ ✓

For divisor $= 9$: $f = 57$, $e = 12 + 64 = 76$ ✓

All three values satisfy $f < 60$ and $f$ is odd.

---

The valid pairs are: $(588, 49)$, $(204, 51)$, $(76, 57)$

Therefore, there are 3 pairs of positive integers $(e, f)$ that satisfy the given conditions.