Find the number of positive integers satisfying the inequality `x^(2) -10x+16lt 0.`

Answer

To solve the inequality \( x^2 - 10x + 16 < 0 \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ x^2 - 10x + 16 < 0 \]

### Step 2: Factor the quadratic expression
To factor the quadratic expression, we look for two numbers that multiply to \( 16 \) (the constant term) and add up to \( -10 \) (the coefficient of \( x \)). The numbers \( -2 \) and \( -8 \) satisfy this condition. Thus, we can factor the expression as:
\[ (x - 2)(x - 8) < 0 \]

### Step 3: Determine the critical points
The critical points occur when the expression equals zero:
\[ (x - 2)(x - 8) = 0 \]
This gives us the points \( x = 2 \) and \( x = 8 \).

### Step 4: Test intervals around the critical points
We will test the intervals defined by the critical points to determine where the product is negative. The intervals to test are:
1. \( (-\infty, 2) \)
2. \( (2, 8) \)
3. \( (8, \infty) \)

- For the interval \( (-\infty, 2) \), choose \( x = 0 \):
\[ (0 - 2)(0 - 8) = ( -2)( -8) = 16 \quad (\text{positive}) \]

- For the interval \( (2, 8) \), choose \( x = 5 \):
\[ (5 - 2)(5 - 8) = (3)( -3) = -9 \quad (\text{negative}) \]

- For the interval \( (8, \infty) \), choose \( x = 9 \):
\[ (9 - 2)(9 - 8) = (7)(1) = 7 \quad (\text{positive}) \]

### Step 5: Identify the solution set
The inequality \( (x - 2)(x - 8) < 0 \) holds true in the interval:
\[ (2, 8) \]

### Step 6: Find the positive integers in the interval
Now, we need to find the positive integers that lie in the interval \( (2, 8) \). The positive integers in this interval are:
\[ 3, 4, 5, 6, 7 \]

### Step 7: Count the positive integers
Counting these integers, we find there are:
\[ 5 \text{ positive integers} \]

### Final Answer
Thus, the number of positive integers satisfying the inequality \( x^2 - 10x + 16 < 0 \) is:
\[ \boxed{5} \]