1. There are two positive integers, one of which is a square number. If the sum of the two numbers is 2006 smaller than their product, find the difference between the two numbers.
(1 mark)現有兩個正整數, 其中一個是平方數。若兩數之和比它們之積小 2006 , 求兩數之差。
1. 666
1. Let the two numbers be $x$ and $y$. Then we have $x+y=x y-2006$. Upon factorisation, we have
$$
(x-1)(y-1)=2007 \text {. }
$$

Since $2007=3^{2} \times 223$, it has 6 positive factors, namely, 1, 3, 9, 223, 669 and 2007 . As one of $x$ and $y$ (say, $x$ ) is a perfect square, $x-1$ must be 1 less than a perfect square. Among $1,3,9,223$, 669 and 2007 , only 3 has this property $\left(3=2^{2}-1\right)$. Therefore we should take $x-1=3$ and $y-1=669$, i.e. $x=4$ and $y=670$. It follows that the answer is $670-4=666$.