Sure! Let's break down the given problem and solve it step-by-step. We need to find how many natural numbers [tex]\( n \)[/tex] make the expression [tex]\(\frac{n^2 - 9n + 27}{n - 7}\)[/tex] equal to a natural number.
### Step 1: Investigate the Expression
We start by examining the expression:
[tex]\[
\frac{n^2 - 9n + 27}{n - 7}
\][/tex]
To simplify this expression, we use polynomial long division or synthetic division. Let's perform polynomial long division:
1. Divide [tex]\( n^2 \)[/tex] by [tex]\( n \)[/tex]:
[tex]\[
n
\][/tex]
2. Multiply [tex]\( n \)[/tex] by [tex]\( n - 7 \)[/tex]:
[tex]\[
n \cdot (n - 7) = n^2 - 7n
\][/tex]
3. Subtract [tex]\( n^2 - 7n \)[/tex] from [tex]\( n^2 - 9n + 27 \)[/tex]:
[tex]\[
(n^2 - 9n + 27) - (n^2 - 7n) = -2n + 27
\][/tex]
4. Divide [tex]\( -2n \)[/tex] by [tex]\( n \)[/tex]:
[tex]\[
-2
\][/tex]
5. Multiply [tex]\( -2 \)[/tex] by [tex]\( n - 7 \)[/tex]:
[tex]\[
-2 \cdot (n - 7) = -2n + 14
\][/tex]
6. Subtract [tex]\( -2n + 14 \)[/tex] from [tex]\( -2n + 27 \)[/tex]:
[tex]\[
(-2n + 27) - (-2n + 14) = 27 - 14 = 13
\][/tex]
Thus, the polynomial division gives us:
[tex]\[
n - 2 + \frac{13}{n - 7}
\][/tex]
### Step 2: Ensure the Result is a Natural Number
For [tex]\(\frac{n^2 - 9n + 27}{n - 7}\)[/tex] to be a natural number, the remainder [tex]\(\frac{13}{n - 7}\)[/tex] must be an integer. This means that [tex]\( n - 7 \)[/tex] must be a divisor of 13. The divisors of 13 are [tex]\( \pm 1 \)[/tex] and [tex]\( \pm 13 \)[/tex].
### Step 3: Determine Valid [tex]\(n\)[/tex]
Solve for [tex]\( n \)[/tex] using these divisors:
1. [tex]\( n - 7 = 1 \)[/tex]:
[tex]\[
n = 8
\][/tex]
2. [tex]\( n - 7 = -1 \)[/tex]:
[tex]\[
n = 6 \quad \text{(not a valid solution since it does not result in a natural number in the simplified expression)}
\][/tex]
3. [tex]\( n - 7 = 13 \)[/tex]:
[tex]\[
n = 20
\][/tex]
4. [tex]\( n - 7 = -13 \)[/tex]:
[tex]\[
n = -6 \quad \text{(not a natural number)}
\][/tex]
### Conclusion
Thus, the natural numbers [tex]\( n \)[/tex] such that [tex]\(\frac{n^2 - 9n + 27}{n - 7}\)[/tex] is a natural number are:
[tex]\[
n = 8, \quad n = 20
\][/tex]
Therefore, there are [tex]\(\boxed{2}\)[/tex] such natural numbers.
