Square of an Integer

For how many integers n is \displaystyle\frac{n}{20-n} the square of an integer?
Source: NCTM Mathematics Teacher

SOLUTION
The square of an integer is a positive integer, therefore \displaystyle\frac{n}{20-n} is a positive integer. If n<0, 20-n<0 or n>20, a contradiction. Thus, n>0. If the numerator is positive, the denominator must also be positive
20-n>0
n<20
Furthermore, the numerator must be greater than or equal to the denominator for the rational number \displaystyle\frac{n}{20-n} to be an integer
n\geqslant 20-n
2n\geqslant 20
n\geqslant 10
Verification
\displaystyle\frac{10}{20-10}=\frac{10}{10}=1^2

\displaystyle\frac{16}{20-16}=\frac{16}{4}=2^2

\displaystyle\frac{18}{20-18}=\frac{18}{2}=3^2

Don’t forget the special value
\displaystyle\frac{0}{20-0}=\frac{0}{20}=0^2

There are 4 values of n=\left\{0,10,16,18\right\} for which \displaystyle\frac{n}{20-n} is the square of an integer.

Answer: 4

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , . Bookmark the permalink.

One Response to Square of an Integer

  1. gaurav maity says:

    for n= 16 the equation would be [16/(20-16)] = 16/4 = 4 = 2^2 which is a perfect square. Hence the ans is 16.

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