Find a common factor of $(127^{127} + 97^{127})$ and $(127^{97} + 97^{97})$.

### Concept & Formula The problem tests the divisibility property of the algebraic sum of powers. The fundamental rule states that any expression in the form $(x^n + a^n)$ is perfectly divisible by $(x + a)$ for all **odd** values of $n$. ### Step-by-Step Solution **Given:** Expression 1: $127^{127} + 97^{127}$ Expression 2: $127^{97} + 97^{97}$ **Calculation / Deduction:** * **Analyze Expression 1:** It is in the form $x^n + a^n$ with $x = 127$, $a = 97$, and $n = 127$. Since the exponent ($127$) is an odd number, it is perfectly divisible by $(x + a)$. $$ 127 + 97 = 224 $$ * So, $224$ is a factor of the first expression. * **Analyze Expression 2:** It is also in the form $x^n + a^n$ with $x = 127$, $a = 97$, and $n = 97$. Since the exponent ($97$) is also an odd number, it is perfectly divisible by $(x + a)$. $$ 127 + 97 = 224 $$ * So, $224$ is a factor of the second expression as well. * Since $224$ divides both expressions, it is their common factor. ### Exam Strategy & Shortcut When you see $(x^{\text{odd}} + y^{\text{odd}})$, immediately calculate $x + y$. Here, $127 + 97 = 224$. Since both expressions feature odd powers over the exact same bases, $224$ is guaranteed to be the common factor. Ignore the different exponents entirely. ### Common Pitfall A trap students fall into is trying to find a relationship between the exponents ($127$ and $97$), perhaps trying to find the HCF of the powers. For the $(x^n + a^n)$ rule, the specific value of the exponent does not matter, only its odd/even parity matters. ### Final Answer Therefore, the correct answer is 224.