There is only one integer n for which the expression
is an integer. Find this value of n and show that there are no others.
Solution.
This month's problem nicely illustrates one way mathematics has changed over the past twenty years. We feature two solutions, the first from twenty years ago by the late W.J. Blundon of the Memorial University of Newfoundland, and the second by Juan Mir Pieras of Spain.
We took the problem from Crux Mathematicorum7:1 (1981), page 31. Blundon's solution exploits a trick that is worth remembering for future use. Using long division we see that the denominator goes into the numerator 2n + 9 times with a remainder involving n, specifically
The remainder can be an integer only when 3n - 5 divides evenly into the numerator. Since 3n - 5 and 3n - 4 differ by 1, they are relatively prime; this means that E(n) can be an integer only if 3n - 5 divides 5n + 4. Now the trick: If 3n - 5 divides 5n + 4, then it would necessarily also divide
3(5n + 4) - 5(3n - 5) = 37
Since 37 is prime we conclude that 3n - 5 = ±1 or 3n - 5 = ±37. We quickly eliminate -1 (implying n = 4/3) and -37 (implying n = -32/3), because n is an integer. When 3n - 5 = 1, n = 2 and E(2) = 7.4, which is not an integer. That leaves 3n - 5 = 37, in which case n = 14 and E(14) = 41. Thus E(n) is an integer if and only if n = 14.
Solution by Juan Mir Pieras.
Today most of us have access to a computer that can quickly compute the value of E(n) for a list of possible values of n (so long as the list is not too long). Consequently, all we have to do is to produce a short list of possible values. To this end, just continue the long division as far as possible:
Here the remainder (that is, the quotient involving n) goes to zero as the absolute value of n gets large. E(n) cannot be an integer when the remainder is less than 1/2, and an easy analysis determines that the remainder fails to be less than 1/2 only for -23 <= n <= 32. (We leave that analysis to the reader. Details can be found in the Spanish solution) So have your computer check those 56 values of E(n). Once again one sees that the only time E(n) is an integer is when n = 14 and E(14) = 41.
