Question

The number of integers $n(<20)$ for which $n^2-3n+3$ is a perfect square is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

Let the given expression be equal to $m^2$
$n^2-3n+3=m^2$
$n^2-m^2=3n-3$
$(n-m)(n+m)=3(n-1)\times 1$
$=3\times (n-1)$
(i) $n+m=1$

$n-m=3(n-1)$

Solving above two equations simultaneously we get,
$2n=3n-2$
$\therefore$ $n=2$
(ii) $n-m=3$
$n+m=n-1$
$2n=n+2$ or $n-m=n-1$
$n=2$ $n+m=3$
$2n=n+2$
$n=2$
Only one value of $n$ $(i.e.,n=2)$ for which $n^2-3n+3$ is a perfect square.

Hence, option C is correct.