Question:

Step 1: Find the roots of the quadratic equation. To solve the inequality, we first find the roots of the corresponding quadratic equation $x^2 - 10x + 16 = 0$. We can factor the quadratic expression: $$x^2 - 10x + 16 = (x - 2)(x - 8) = 0$$ Setting each factor to zero gives the roots: $$x - 2 = 0 \implies x = 2$$ $$x - 8 = 0 \implies x = 8$$ Thus, the roots are $x = 2$ and $x = 8$. Step 2: Determine the interval where the quadratic expression is negative. The given inequality is $x^2 - 10x + 16 < 0$. Since the coefficient of $x^2$ is positive (it is 1), the parabola opens upwards. For a quadratic expression $ax^2 + bx + c$ with $a > 0$, the expression is negative between its roots. Therefore, $x^2 - 10x + 16 < 0$ when $x$ is strictly between the roots $2$ and $8$. This implies that the inequality is satisfied for $2 < x < 8$. Step 3: Identify the positive integers satisfying the inequality. We need to find the positive integers $x$ that lie in the interval $(2, 8)$. The integers strictly greater than 2 and strictly less than 8 are: $$3, 4, 5, 6, 7$$ All these integers are positive. Step 4: Count the number of such positive integers. Counting the integers identified in the previous step, we have: There are $5$ such positive integers: $3, 4, 5, 6, 7$. The final answer is $\boxed{5}$.