## 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: $4$

Retired high school math teacher.
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### 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.