How many pairs of positive integers e, f satisfy 1/e + 4/f = 1/12 where f is an odd integer less than 60?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3
The equation is solved for \(e\) in terms of \(f\). Since \(e\) must be a positive integer, only the values of \(f\) that make \(e\) positive and a whole number give a pair.
Writing \(e\) in terms of \(f\):
\(\displaystyle \frac{1}{e}=\frac{1}{12}-\frac{4}{f}=\frac{f-48}{12f}\)
So \(\displaystyle e=\frac{12f}{f-48}\)
The possible values of \(f\) (\(e\) is positive only if \(f>48\)):
With \(f\) odd and less than 60, \(f\) can be 49, 51, 53, 55, 57 or 59
Testing each value (\(e\) must be a whole number):
\(f=49\): \(e=\frac{588}{1}=588\), a whole number
\(f=51\): \(e=\frac{612}{3}=204\), a whole number
\(f=53\): \(e=\frac{636}{5}\), not a whole number
\(f=55\): \(e=\frac{660}{7}\), not a whole number
\(f=57\): \(e=\frac{684}{9}=76\), a whole number
\(f=59\): \(e=\frac{708}{11}\), not a whole number
So there are 3 pairs: (588, 49), (204, 51) and (76, 57).