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For every natural number n, the expression 2·7ⁿ + 3·5ⁿ − 5 is divisible by which of the following?

1. 24, for all n ∈ N
2. 21, for all n ∈ N
3. 35, for all n ∈ N
4. 50, for all n ∈ N

Correct answer: 24, for all n ∈ N

Solution

The expression can be simplified and analyzed for divisibility by 24, which is the product of 8 and 3. By checking the expression modulo 8 and modulo 3, we find that it consistently yields results that are divisible by both, confirming that the entire expression is divisible by 24 for all natural numbers n.

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