Discriminating variable

Let the blank number be $c$ . Then, the equation is rewritten as $x^2 - 10x + c = 0$ .

Using the quadratic formula, we have roots $x = \dfrac{10 \pm \sqrt{100 - 4c}}{2} = \dfrac{10 \pm 2\sqrt{25-c}}{2} = 5 \pm \sqrt{25-c}$

For the roots to be positive integers, we must have the discriminant to be nonnegative and a perfect square: $D = 10^2 - 4(1)(c) = 100 - 4c = 4(25-c)$ . Since $4$ is a perfect square, we only need to ensure that $25-c$ is a perfect square. Since $c > 0$ , we only have 5 options for $c:\,9, 16, 21, 24, 25$ .

Then, we need to make sure that $x$ is positive. Substituting these 5 possible choices of $c$ , we see that all 5 of them produces positive values of $x$ .

Hence, we have $\boxed{5}$ choices of $c$ :

• $c= 9$ , which gives us roots $x = 9, 1$
• $c= 16$ , which gives us roots $x = 8, 2$
• $c= 21$ , which gives us roots $x = 7, 3$
• $c= 24$ , which gives us roots $x = 6, 4$
• $c = 25$ , which gives us a repeated root $x= 5$