Key Concepts and Formulas

• Diophantine Equations: Solving linear equations of the form $ax + by = c$ for integer solutions, often with the constraint that the solutions are natural numbers.
• Properties of Complex Numbers: Understanding the cyclic nature of powers of $i$, where $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. This cycle repeats every four powers.
• Factorials and Modular Arithmetic: Understanding that for $n \ge 4$, $n!$ is divisible by 4, meaning $n! \equiv 0 \pmod{4}$.

Step-by-Step Solution

Step 1: Determine the number of elements ($m$) in the set R

The set $R$ is defined as $R = \{(a, b) : a + 5b = 42, a, b \in \mathbb{N}\}$. We need to find the number of pairs $(a, b)$ of natural numbers that satisfy the equation $a + 5b = 42$.

Step 2: Express $a$ in terms of $b$

We rewrite the equation to express $a$ as a function of $b$: $a = 42 - 5b$

Step 3: Apply the natural number constraint for $a$

Since $a \in \mathbb{N}$, we must have $a \ge 1$. Substituting the expression for $a$, we get: $42 - 5b \ge 1$ $41 \ge 5b$ $b \le \frac{41}{5}$ $b \le 8.2$

Step 4: Apply the natural number constraint for $b$

Since $b \in \mathbb{N}$, we must have $b \ge 1$.

Step 5: Find all possible integer values for $b$

Combining the constraints $b \ge 1$ and $b \le 8.2$, the possible integer values for $b$ are $\{1, 2, 3, 4, 5, 6, 7, 8\}$.

Step 6: Calculate the corresponding $a$ values for each $b$

We iterate through each valid $b$ and find the corresponding $a$:

• If $b = 1$, $a = 42 - 5(1) = 37$. So, $(37, 1) \in R$.
• If $b = 2$, $a = 42 - 5(2) = 32$. So, $(32, 2) \in R$.
• If $b = 3$, $a = 42 - 5(3) = 27$. So, $(27, 3) \in R$.
• If $b = 4$, $a = 42 - 5(4) = 22$. So, $(22, 4) \in R$.
• If $b = 5$, $a = 42 - 5(5) = 17$. So, $(17, 5) \in R$.
• If $b = 6$, $a = 42 - 5(6) = 12$. So, $(12, 6) \in R$.
• If $b = 7$, $a = 42 - 5(7) = 7$. So, $(7, 7) \in R$.
• If $b = 8$, $a = 42 - 5(8) = 2$. So, $(2, 8) \in R$.

Since all pairs have natural numbers for both $a$ and $b$, the set $R$ is $\{(37, 1), (32, 2), (27, 3), (22, 4), (17, 5), (12, 6), (7, 7), (2, 8)\}$.

Step 7: Determine $m$

The number of elements in set $R$ is $m = 8$.

Step 8: Evaluate the summation $\sum_{n=1}^m (1 - i^{n!})$

We need to evaluate the sum $\sum_{n=1}^8 (1 - i^{n!}) = x + iy$. We can split the sum: $\sum_{n=1}^8 (1 - i^{n!}) = \sum_{n=1}^8 1 - \sum_{n=1}^8 i^{n!}$

Step 9: Calculate $\sum_{n=1}^8 1$

The first part is $\sum_{n=1}^8 1 = 1 \times 8 = 8$.

Step 10: Analyze $i^{n!}$ for each value of $n$ from 1 to 8

The value of $i^k$ depends on $k \pmod{4}$.

• For $n = 1$: $n! = 1! = 1$. $i^{1!} = i^1 = i$.
• For $n = 2$: $n! = 2! = 2$. $i^{2!} = i^2 = -1$.
• For $n = 3$: $n! = 3! = 6$. $6 \pmod{4} \equiv 2$. $i^{3!} = i^6 = i^2 = -1$.
• For $n = 4$: $n! = 4! = 24$. $24 \pmod{4} \equiv 0$. $i^{4!} = i^{24} = 1$.
• For $n \ge 4$: For any integer $n \ge 4$, the term $n!$ will always contain $4$ as a factor. This means $n!$ is always a multiple of $4$. Therefore, for $n = 4, 5, 6, 7, 8$, $n! \equiv 0 \pmod{4}$, which implies $i^{n!} = 1$.

Step 11: Calculate the sum $\sum_{n=1}^8 i^{n!}$

$\sum_{n=1}^8 i^{n!} = i^{1!} + i^{2!} + i^{3!} + i^{4!} + i^{5!} + i^{6!} + i^{7!} + i^{8!}$ $= i + (-1) + (-1) + 1 + 1 + 1 + 1 + 1$ $= i - 2 + 5$ $= 3 + i$

Step 12: Calculate the total sum $\sum_{n=1}^8 (1 - i^{n!})$

$\sum_{n=1}^8 (1 - i^{n!}) = \left( \sum_{n=1}^8 1 \right) - \left( \sum_{n=1}^8 i^{n!} \right)$ $= 8 - (3 + i)$ $= 8 - 3 - i$ $= 5 - i$

Step 13: Identify $x$ and $y$

We have $\sum_{n=1}^m (1 - i^{n!}) = x + iy$. Comparing $5 - i$ with $x + iy$, we get $x = 5$ and $y = -1$.

Step 14: Calculate $m + x + y$

Substitute the values we found: $m = 8$, $x = 5$, and $y = -1$. $m + x + y = 8 + 5 + (-1)$ $m + x + y = 13 - 1$ $m + x + y = 12$

Common Mistakes & Tips

• Natural Number Definition: Always remember that natural numbers start from 1 (i.e., $\mathbb{N} = \{1, 2, 3, ...\}$).
• Powers of i: Be careful when evaluating powers of $i$. Remember the cycle: $i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$. Also, remember to use modulo 4.

Summary

By carefully analyzing the Diophantine equation and the properties of complex numbers, we found $m=8$, $x=5$, and $y=-1$. Therefore, $m+x+y = 12$.

The final answer is \boxed{12}, which corresponds to option (A).