Directions: Solve the number series to answer the following questions. $\sqrt[n]{A}$, 31, B, C, 439, B + 687, 1296 Note: (i) All the terms in the given series and 'n' are positive integer. (ii) Difference between 31 and B is X, which has 3 factors excluding X itself, those are 13, 5 and (A - 8). What is the value of $(A + C - n)$?

### Concept & Number Series Pattern Finding the unknown variables requires analyzing proper factors of a number and identifying algebraic progression patterns based on differences between terms. ### Step-by-Step Solution 1. **Determine $X$ and $A$:** $X$ has exactly 3 proper factors (factors excluding the number itself): 13, 5, and $(A - 8)$. Since 13 and 5 are prime, their product is $13 \times 5 = 65$. The proper factors of 65 are 1, 5, and 13. This gives $X = 65$. The third factor must be 1, so $A - 8 = 1 \Rightarrow A = 9$. 2. **Determine $B$:** The difference between 31 and $B$ is $X = 65$. Since series terms are increasing based on the progression, $B = 31 + 65 = 96$. 3. **Determine $n$:** The first term is $\sqrt[n]{A} = \sqrt[n]{9}$. For this to be a positive integer, $n = 2$, making the first term 3. 4. **Identify the Series Pattern:** The series is 3, 31, 96, $C$, 439, $(96 + 687 = 783)$, 1296. Let's calculate the differences between consecutive terms: $31 - 3 = 28$; $96 - 31 = 65$. 5. The differences (28, 65) follow the pattern $k^3 + 1$: $3^3 + 1 = 28$, $4^3 + 1 = 65$. 6. **Determine $C$:** The next difference should be $5^3 + 1 = 126$. So, $C = 96 + 126 = 222$. (Verify next term: $222 + (6^3 + 1) = 222 + 217 = 439$, which perfectly matches the given series). 7. **Calculate Final Expression:** $(A + C - n) = (9 + 222 - 2) = 229$. ### Exam Strategy & Shortcut When told a number has exactly three proper factors and two are prime, recognize immediately that the number is the product of those distinct primes and the third factor MUST be 1. This instantly solves for $A$ without trial and error. ### Common Pitfall Assuming the 3 factors could imply a larger composite number, which leads to overcomplicating $(A - 8)$. Remember that 1 is always a proper factor of any positive integer. ### Final Answer Therefore, the correct answer is **229**.