Divide by 12

Learning Objectives Define modular arithmetic and congruence modulo 12. Translate word problems involving 12-hour cycles into systems of linear congruences. Solve systems of two linear congruences (mod 12) using substitution and elimination methods. Find and correctly notate the complete set of integer solutions for a system of congruences. Identify when a modular inverse is needed and calculate it for modulus 12. Verify solutions within the modular system by checking their congruence. If a meeting starts at 9 AM and lasts for 40 hours, what time does it end? 🕰️ This clock puzzle is a system of equations problem where we 'divide by 12' to find the answer! In this tutorial, we will explore a fascinating application of systems of equations within modular arithmetic,...

TermDefinitionExample Modular ArithmeticA system of arithmetic for integers where numbers 'wrap around' upon reaching a certain value, the modulus. It is often called 'clock arithmetic' because it models the 12 hours on a clock.In modulo 12, 14 o'clock is the same as 2 o'clock, because 14 has a remainder of 2 when divided by 12. Congruence Modulo 12Two integers, 'a' and 'b', are said to be congruent modulo 12 if they have the same remainder when divided by 12. This is written as a ≡ b (mod 12).26 ≡ 2 (mod 12) because both 26 and 2 have a remainder of 2 when divided by 12. Also, 26 - 2 = 24, which is a multiple of 12. Congruence ClassThe set of all integers that are congruent to a specific integer modulo 12. Each class is represented by its...

Definition of Congruence a \equiv b \pmod{n} \iff n | (a-b) This is the formal definition. Two numbers 'a' and 'b' are congruent modulo 'n' if their difference (a-b) is evenly divisible by 'n'. For this lesson, n=12. Properties of Modular Arithmetic If a \equiv b \pmod{n} and c \equiv d \pmod{n}, then: 1. a+c \equiv b+d \pmod{n} 2. a-c \equiv b-d \pmod{n} 3. ac \equiv bd \pmod{n} These properties allow us to perform standard algebraic operations like adding, subtracting, and multiplying equations within a system of congruences, just as we do with regular equations. Solving ax ≡ b (mod 12) If \text{gcd}(a, 12) = 1, then x \equiv b \cdot a^{-1} \pmod{12} To solve for x, you can't simply divide by 'a'. Instead, y...