Question: What is the remainder when $ 7^5 + 7^6 $ is divided by 6?
["# Remainder When $ 7^5 + 7^6 $ Is Divided by 6: A Simple Modular Arithmetic Guide", "When faced with the problem of finding the remainder when $ 7^5 + 7^6 $ is divided by 6, many students wonder how to simplify the expression efficiently using modular arithmetic. This article explains step-by-step how to solve the question: What is the remainder when $ 7^5 + 7^6 $ is divided by 6? We’ll use modular arithmetic — a powerful tool in number theory — to break this down in a clear and intuitive way.", "---", "## Understanding the Problem", "We are asked to compute:\n$$\n(7^5 + 7^6) \mod 6\n$$\nInstead of calculating large powers directly, we use modular arithmetic properties to simplify the expression. The goal is to find the remainder after dividing $ 7^5 + 7^6 $ by 6.", "---", "## Step 1: Simplify the Base Modulo 6", "First, reduce the base 7 modulo 6:\n$$\n7 \mod 6 = 1\n$$\nSo,\n$$\n7 \equiv 1 \pmod{6}\n$$", "---", "## Step 2: Apply Modular Equivalence to Powers", "Using the fact that if $ a \equiv b \pmod{n} $, then $ a^k \equiv b^k \pmod{n} $, we substitute:\n$$\n7^5 \equiv 1^5 \pmod{6} \\n7^6 \equiv 1^6 \pmod{6}\n$$", "Since $ 1^k = 1 $ for any positive integer $ k $,\n$$\n7^5 \equiv 1 \pmod{6} \\n7^6 \equiv 1 \pmod{6}\n$$", "---", "## Step 3: Add the Results", "Now compute the sum modulo 6:\n$$\n7^5 + 7^6 \equiv 1 + 1 = 2 \pmod{6}\n$$", "---", "## Step 4: Final Answer", "Therefore, the remainder when $ 7^5 + 7^6 $ is divided by 6 is:\n$$\n\boxed{2}\n$$", "This elegant solution shows how modular arithmetic transforms a seemingly complex exponent problem into a straightforward calculation.", "---", "## Why This Method Works", "By recognizing that powers of a number congruent to 1 mod 6 remain congruent to 1, we avoid computing large exponents. This technique is widely used in number theory, cryptography, and computer science for efficient modular computations.", "---", "Try Practice: Test with smaller powers, like $ 7^2 + 7^3 \mod 6 $, to reinforce your understanding.\nThis method guarantees accurate results with minimal computation — ideal for both students and problem solvers alike!"]
