We want \( 80k + 16 \equiv 76 \pmod{100} \), so:

["# Solving the Modular Equation: ( 80k + 16 \equiv 76 \pmod{100} )", "Modular arithmetic plays an essential role in number theory, cryptography, computer science, and many algorithmic problems. One common type of problem involves solving linear congruences such as:", "[\n80k + 16 \equiv 76 \pmod{100}\n]", "If you want to find integer solutions for ( k ), this article guides you step-by-step through solving this modular equation.", "## Step 1: Simplify the Congruence", "Start with the original equation:", "[\n80k + 16 \equiv 76 \pmod{100}\n]", "Subtract 16 from both sides:", "[\n80k \equiv 60 \pmod{100}\n]", "Now we aim to solve:", "[\n80k \equiv 60 \pmod{100}\n]", "## Step 2: Simplify Using the Greatest Common Divisor (GCD)", "The key to solving ( ax \equiv b \pmod{m} ) is to divide through by ( \gcd(a, m) ). Here, ( a = 80 ), ( b = 60 ), and ( m = 100 ).", "Calculate ( d = \gcd(80, 100) ):", "[\nd = \gcd(80, 100) = 20\n]", "Check if ( d \mid b ):", "[\n20 \mid 60 \quad \ ext{(since } 60 \div 20 = 3\ ext{)}\n]", "Because the gcd divides the right-hand side, solutions exist. Divide the entire congruence by 20:", "[\n\frac{80}{20}k \equiv \frac{60}{20} \pmod{\frac{100}{20}} \quad \Rightarrow \quad 4k \equiv 3 \pmod{5}\n]", "## Step 3: Solve the Reduced Congruence", "Now solve:", "[\n4k \equiv 3 \pmod{5}\n]", "To isolate ( k ), multiply both sides by the modular inverse of 4 modulo 5. Find ( 4^{-1} \mod 5 ):", "Try small values:\n( 4 \ imes 4 = 16 \equiv 1 \pmod{5} \Rightarrow 4^{-1} \equiv 4 \pmod{5} )", "Multiply both sides of ( 4k \equiv 3 \pmod{5} ) by 4:", "[\nk \equiv 4 \ imes 3 = 12 \equiv 2 \pmod{5}\n]", "Thus,", "[\nk \equiv 2 \pmod{5}\n]", "## Step 4: Express All Solutions Modulo 100", "The solutions to ( k \equiv 2 \pmod{5} ) are all integers ( k = 5t + 2 ) for integer ( t ).", "But we divided the original modulus by 20, so we must lift this solution back to modulo 100.", "We seek all solutions modulo 100 that satisfy:", "[\nk \equiv 2 \pmod{5}\n]", "The complete set of solutions modulo 100 is:", "[\nk \equiv 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97 \pmod{100}\n]", "These are all integers ( k ) such that ( k \mod 5 = 2 ), within the range 0 to 99.", "## Step 5: Final Answer", "The solutions to the congruence ( 80k + 16 \equiv 76 \pmod{100} ) are all integers ( k ) congruent to 2 modulo 5. Explicitly, they are:", "[\n\boxed{k \equiv 2 \pmod{5}, \quad \ ext{with } k \in { 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97 } \ ext{ modulo } 100}\n]", "## Why This Matters", "Modular equations like these appear in algorithmic fairness, cryptographic protocols, and error-checking systems. Understanding how to solve such congruences is crucial for experts developing secure systems and efficient computational methods.", "---", "For a deeper understanding of modular arithmetic in real-world applications, see resources on number theory fundamentals and computational cryptography.", "---", "Keywords: modular equation, solve 80k + 16 ≡ 76 mod 100, solve linear congruence, computer science modular arithmetic, number theory solved example, mathematical solution guide\nMeta description: Solve ( 80k + 16 \equiv 76 \pmod{100} ) with step-by-step explanation. Learn how to find integer ( k ) using GCD, modular inverses, and lifting solutions. Ideal for math students, cryptographers, and software engineers."]