5Question: What is the remainder when the sum of the sequence starting at $11071$ with a common difference of $2$ up to $5$ terms is divided by $8$?

["Title: How to Find the Remainder When a Sequence’s Sum Is Divided by 8 – A Step-by-Step Guide Using 11071", "Meta Description: Learn how to determine the remainder when the sum of a 5-term arithmetic sequence starting at 11071 with a common difference of 2 is divided by 8. This easy math method helps solve real-world problems with modular arithmetic.", "---", "## Understanding the Problem: Sum of a 5-Term Sequence Starting at 11071 with Difference 2", "The sequence begins at 11071 and increases by 2 for each term, forming an arithmetic sequence:\n[\n11071,\ 11073,\ 11075,\ 11077,\ 11079\n]\nWe need to compute the sum of these five terms and then find the remainder when divided by 8.", "---", "### Step 1: Compute the Sum of the Sequence", "One quick way to find the sum of an arithmetic sequence is:\n[\n\ ext{Sum} = \frac{n}{2} \ imes (\ ext{first term} + \ ext{last term})\n]\nHere, ( n = 5 ), first term ( a_1 = 11071 ), last term ( a_5 = 11079 ).\n[\n\ ext{Sum} = \frac{5}{2} \ imes (11071 + 11079) = \frac{5}{2} \ imes 22150 = 5 \ imes 11075 = 55375\n]", "So, the sum is 55375.", "---", "### Step 2: Find the Remainder When Divided by 8", "Now, compute:\n[\n55375 \div 8 \quad \ ext{and find } 55375 \mod 8\n]", "Instead of full division, use modular arithmetic to simplify. First, reduce the first term modulo 8 and leverage the common difference.", "Note: The sequence increases by 2, and we’re summing 5 terms, so modular behavior repeats every 4 terms due to the cycle of even increments mod 8.", "But easier: compute ( 11071 \mod 8 ) and proceed step-by-step.", "#### Find ( 11071 \mod 8 ):\nDivide 11071 by 8:\n[\n8 \ imes 1383 = 11064 \quad \Rightarrow \quad 11071 - 11064 = 7\n]\nSo,\n[\n11071 \equiv 7 \pmod{8}\n]", "Since the sequence increases by 2 each time:\n- Term 1: 7 mod 8\n- Term 2: (7 + 2 = 9 \equiv 1 \pmod{8})\n- Term 3: (1 + 2 = 3 \pmod{8})\n- Term 4: (3 + 2 = 5 \pmod{8})\n- Term 5: (5 + 2 = 7 \pmod{8})", "Now sum the residues:\n[\n7 + 1 + 3 + 5 + 7 = 23\n]\nNow compute ( 23 \mod 8 ):\n[\n8 \ imes 2 = 16, \quad 23 - 16 = 7 \quad \Rightarrow \quad 23 \equiv 7 \pmod{8}\n]", "Thus, the remainder when the sum is divided by 8 is 7.", "---", "### Alternative Quick Check: Sum Modulo 8 Directly", "[\n\ ext{Sum} = 55375\n]\nCheck:\n( 55375 \div 8 ) — instead of full division, sum residues mod 8:\n[\n55375 \equiv 7 \pmod{8}\n]\nMatches earlier result.", "---", "## Why This Method Works", "Working with modular residues instead of large numbers simplifies calculations. Since each term is obtained by adding 2 mod 8, and the sequence cycles predictable values (7, 1, 3, 5, 7), summing their residues avoids lengthy division.", "---", "## Practical Takeaway: Remainders Matter", "Understanding remainders helps in scheduling, coding, cryptography, and error checking. Modular arithmetic provides efficient ways to handle large sums — especially common in computer science and digital systems.", "---", "### Final Answer", "The remainder when the sum of the sequence starting at 11071 with a common difference of 2 (up to 5 terms) is divided by 8 is 7.", "---", "Keywords for SEO:\nremainder when divided by 8, sum of arithmetic sequence mod 8, modular arithmetic example, find remainder 11071 sum 2 difference 5 terms, math problem solution with modulo, remainder calc step-by-step", "Related Articles:\n- How to calculate remainders using modular arithmetic\n- Arithmetic sequences and their sum formulas\n- Practical applications of remainders in coding and math", "---\nNote: Use modular reduction early to simplify complex sums. This method is efficient, accurate, and widely applicable."]