Find the number of all positive integers n, such that \frac{{{3^{n + 1}} + {7^{n + 1}}}}{{{3^{n - 1}} + {7^{n - 1}}}}

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We are tasked with finding the number of all positive integers \( n \) such that:

\[
\frac{{3^{n+1} + 7^{n+1}}}{{3^{n-1} + 7^{n-1}}}
\]

### Step 1: Simplify the expression

First, rewrite the given expression:

\[
\frac{{3^{n+1} + 7^{n+1}}}{{3^{n-1} + 7^{n-1}}}
= \frac{{3 \cdot 3^n + 7 \cdot 7^n}}{{\frac{{3^n}}{{3}} + \frac{{7^n}}{{7}}}} 
= \frac{{3 \cdot 3^n + 7 \cdot 7^n}}{{\frac{3^n}{3} + \frac{7^n}{7}}}
\]

### Step 2: Investigate specific values of \( n \)

We can now test values of \( n \).

- For \( n = 1 \):

\[
\frac{{3^{1+1} + 7^{1+1}}}{{3^{1-1} + 7^{1-1}}}
= \frac{{3^2 + 7^2}}{{3^0 + 7^0}} = \frac{{9 + 49}}{{1 + 1}} = \frac{58}{2} = 29
\]

- For \( n = 2 \):

\[
\frac{{3^{2+1} + 7^{2+1}}}{{3^{2-1} + 7^{2-1}}}
= \frac{{3^3 + 7^3}}{{3^1 + 7^1}} = \frac{{27 + 343}}{{3 + 7}} = \frac{370}{10} = 37
\]

- For \( n = 3 \):

\[
\frac{{3^{3+1} + 7^{3+1}}}{{3^{3-1} + 7^{3-1}}}
= \frac{{3^4 + 7^4}}{{3^2 + 7^2}} = \frac{{81 + 2401}}{{9 + 49}} = \frac{2482}{58} = 42.79 \quad (\text{not an integer})
\]

Thus, the expression only yields integers for \( n = 1 \) and \( n = 2 \).

### Conclusion

The number of positive integers \( n \) for which the expression is an integer is \( 2 \).

### Answer: \( \boxed{2} \)

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Would you like any details or further clarification?

Here are 5 questions to expand on the topic:

1. How would you approach this problem if higher values of \( n \) were needed?
2. Can this method be applied to other similar exponential expressions?
3. What are the general techniques for simplifying complex algebraic fractions?
4. How would this change if we were working in modular arithmetic?
5. Can this method be extended to non-integer exponents?

**Tip**: When dealing with exponents, always check small values first—they often reveal patterns or solutions quickly.

Find the number of all positive integers n, such that \frac{{3^{n + 1}} + {7^{n + 1}}}{{3^{n - 1}} + {7^{n - 1}}}.

Solve the exponential equation \frac{{3^{n+1} + 7^{n+1}}}{{3^{n-1} + 7^{n-1}}} to find the number of positive integer values for n. This detailed solution explores algebraic techniques and simplification of rational fractions, suitable for students in Grades 9-12.

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