\hspace{-16}$The no. of positive integer value of $\bf{n}$ for which $\bf{n^2 - 19n + 99}$\\\\ is perfect square.
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1 Answers
2305n2 - 20n + 100 = (n-10)2 + n - 1
Now we try to find the value of n for which the given expression lies between (n-9)2 and (n-10)2
(n-9)2>(n-10)2 + n - 1> (n-10)2
or,
(n-10)2 + 2n -19>(n-10)2 + n - 1>(n-10)2
Therefore,
2n -19>n - 1
or n>18
Hence for n>18 the given expression lies between 2 perfect squares i.e. it cannot be a perfect square.
For n<=18
Answers are 1,9,10,18.
