One less than $(49)^{15}$ is exactly divisible by

### Concept & Formula This question tests the divisibility rules of algebraic polynomials. For any positive integer $n$, the difference of two $n$-th powers, $x^n - y^n$, is always perfectly divisible by $(x - y)$. $$ \frac{x^n - y^n}{x - y} = \text{Integer} $$ ### Step-by-Step Solution * **Given:** The mathematical expression for "One less than $(49)^{15}$" is $(49^{15} - 1)$. * **Format to match identity:** We can rewrite the constant $1$ as $1^{15}$ without changing its value. $$49^{15} - 1^{15}$$ * **Apply the divisibility rule:** Since $x^n - y^n$ is divisible by $(x - y)$, substitute $x = 49$ and $y = 1$. The expression is divisible by $(49 - 1) = 48$. * **Find factors of 48:** If a massive number is exactly divisible by $48$, it must also be exactly divisible by all the fundamental factors of $48$. Factors of $48$: $1, 2, 3, 4, 6, 8, 12, 16, 24, 48$. * **Deduction:** Check the given options ($8, 14, 50, 51$). The only option that is a factor of $48$ is $8$. ### Exam Strategy & Shortcut When asked for a static divisor of an algebraic expression, substituting $n = 1$ often reveals the core factor. Let the exponent be $1$. $49^1 - 1 = 48$. Scan the options for a number that perfectly divides $48$. Only $8$ works. ### Common Pitfall Seeing $(49)^{15} - 1$, a student might associate $49$ with $7^2$ and mistakenly guess that $14$ is a divisor, or falsely apply the $x^n + y^n$ rule to assume it is divisible by $(49 + 1) = 50$. Strictly follow the minus sign rule. ### Final Answer Therefore, the correct answer is **8**.