Key Concepts and Formulas

This problem involves finding the sum of numbers that form arithmetic progressions (APs). The key formulas required are:

1. General term of an AP: For an AP with first term $a_1$, common difference $d$, and $n$ terms, the $n$-th term is $a_n = a_1 + (n-1)d$.
2. Sum of an AP: The sum of the first $n$ terms of an AP is $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$.
3. Divisibility and Remainders: A number $N$ that leaves a remainder $R$ when divided by $D$ can be expressed as $N = Dk + R$, where $k$ is an integer.

Step-by-Step Solution

The problem asks for the sum of all two-digit positive numbers (from 10 to 99) that, when divided by 7, yield a remainder of 2 or 5. We will solve this by finding the sum of numbers for each remainder separately and then adding these sums.

Part 1: Sum of two-digit numbers with a remainder of 2 when divided by 7.

These numbers are of the form $7k+2$.

• Step 1.1: Find the first two-digit number of the form $7k+2$. We need $7k+2 \ge 10$. $7k \ge 8$ $k \ge \frac{8}{7}$ Since $k$ must be an integer, the smallest possible value for $k$ is 2. The first term ($a_1$) is $7(2) + 2 = 14 + 2 = 16$.

• Step 1.2: Find the last two-digit number of the form $7k+2$. We need $7k+2 \le 99$. $7k \le 97$ $k \le \frac{97}{7} \approx 13.85$ Since $k$ must be an integer, the largest possible value for $k$ is 13. The last term ($a_n$) is $7(13) + 2 = 91 + 2 = 93$.

• Step 1.3: Find the number of terms in this AP. The sequence is an AP with $a_1 = 16$, $a_n = 93$, and a common difference $d=7$ (as $k$ increases by 1, the term increases by 7). Using the formula $a_n = a_1 + (n-1)d$: $93 = 16 + (n-1)7$ $77 = (n-1)7$ $11 = n-1$ $n = 12$. Alternatively, the values of $k$ range from 2 to 13, so there are $13 - 2 + 1 = 12$ terms.

• Step 1.4: Calculate the sum of this AP. Using the sum formula $S_n = \frac{n}{2}(a_1 + a_n)$: $S_{12} = \frac{12}{2}(16 + 93)$ $S_{12} = 6(109)$ $S_{12} = 654$.

Part 2: Sum of two-digit numbers with a remainder of 5 when divided by 7.

These numbers are of the form $7k+5$.

• Step 2.1: Find the first two-digit number of the form $7k+5$. We need $7k+5 \ge 10$. $7k \ge 5$ $k \ge \frac{5}{7}$ Since $k$ must be an integer, the smallest possible value for $k$ is 1. The first term ($a_1$) is $7(1) + 5 = 7 + 5 = 12$.

• Step 2.2: Find the last two-digit number of the form $7k+5$. We need $7k+5 \le 99$. $7k \le 94$ $k \le \frac{94}{7} \approx 13.42$ Since $k$ must be an integer, the largest possible value for $k$ is 13. The last term ($a_n$) is $7(13) + 5 = 91 + 5 = 96$.

• Step 2.3: Find the number of terms in this AP. The sequence is an AP with $a_1 = 12$, $a_n = 96$, and a common difference $d=7$. Using the formula $a_n = a_1 + (n-1)d$: $96 = 12 + (n-1)7$ $84 = (n-1)7$ $12 = n-1$ $n = 13$. Alternatively, the values of $k$ range from 1 to 13, so there are $13 - 1 + 1 = 13$ terms.

• Step 2.4: Calculate the sum of this AP. Using the sum formula $S_n = \frac{n}{2}(a_1 + a_n)$: $S_{13} = \frac{13}{2}(12 + 96)$ $S_{13} = \frac{13}{2}(108)$ $S_{13} = 13 \times 54$ $S_{13} = 702$.

Part 3: Calculate the total sum.

The total sum is the sum of the numbers from Part 1 and Part 2, as these are distinct sets of numbers. Total Sum = (Sum of numbers with remainder 2) + (Sum of numbers with remainder 5) Total Sum = $654 + 702 = 1356$.

Common Mistakes & Tips

• Range Check: Ensure that the first and last terms identified are indeed two-digit numbers (between 10 and 99 inclusive).
• Integer Values of k: Carefully determine the smallest and largest integer values for $k$ that satisfy the inequalities for the first and last terms.
• Counting Terms: When determining the number of terms ($n$), remember that if $k$ ranges from $k_{min}$ to $k_{max}$, the number of terms is $k_{max} - k_{min} + 1$.
• Separate APs: Treat the numbers with remainder 2 and remainder 5 as forming two independent arithmetic progressions.

Summary

We identified two distinct arithmetic progressions based on the given remainder conditions. For each progression, we found the first term, last term, and the number of terms within the two-digit range. We then applied the formula for the sum of an arithmetic progression to find the sum for each case. Finally, we added these two sums to obtain the total sum of all qualifying two-digit numbers. The sum of two-digit numbers yielding a remainder of 2 when divided by 7 is 654, and the sum of two-digit numbers yielding a remainder of 5 when divided by 7 is 702. The total sum is $654 + 702 = 1356$.

The final answer is $\boxed{1356}$ which corresponds to option (A).