Key Concepts:
Modulo Operation:
The modulo operation finds the remainder when one integer is divided by another. For example, \(11 \equiv 2 \,(\text{mod } 3)\), because 11 gives the remainder 2 when \(8\) is divided by \(3\) (\(\frac{11}{3} = 3 + \frac{2}{3}\)). \\Formally, \(a \equiv r \,(\text{mod } m)\) if \(a = qm + r\), where \(q\) is the quotient and \(0 \leq r < m\).Congruence Modulo:
Two integers \(a\) and \(b\) are congruent modulo \(m\), written as \(a \equiv b \,(\text{mod } m)\), if their difference is divisible by \(m\). Formally: \[ a \equiv b \,(\text{mod } m) \iff m \,|\, (a - b). \]Multiplication Table Modulo \(m\):
A table showing the products of integers under modulo \(m\), useful for understanding modular arithmetic operations.Example 1: Verify if \(17 \equiv 5 \,(\text{mod } 6)\).
Solution:
Check if \(17 - 5\) is divisible by 6: \[ 17 - 5 = 12. \] Since \(12\) is divisible by \(6\), we conclude: \[ 17 \equiv 5 \,(\text{mod } 6). \]Example 2: Compute \((7 + 10) \,(\text{mod } 5)\).
Solution:
Add the numbers: \[ 7 + 10 = 17. \] Find the remainder when \(17\) is divided by \(5\): \[ 17 \div 5 = 3 \text{ remainder } 2. \] Therefore: \[ (7 + 10) \,(\text{mod } 5) = 2. \]Example 3: Compute \((9 \cdot 8) \,(\text{mod } 7)\).
Solution:
Multiply the numbers: \[ 9 \cdot 8 = 72. \] Find the remainder when \(72\) is divided by \(7\): \[ 72 \div 7 = 10 \text{ remainder } 2. \] Therefore: \[ (9 \cdot 8) \,(\text{mod } 7) = 2. \]Example 4: Compute \(3^4 \,(\text{mod } 5)\).
Solution:
Calculate the power: \[ 3^4 = 81. \] Find the remainder when \(81\) is divided by \(5\): \[ 81 \div 5 = 16 \text{ remainder } 1. \] Therefore: \[ 3^4 \,(\text{mod } 5) = 1. \]Example 5: Multiplication Table Modulo \(4\).
Solution:
Construct the table for integers \(0, 1, 2, 3\) modulo \(4\): \[ \begin{array}{c|cccc} \cdot \,(\text{mod } 4) & 0 & 1 & 2 & 3 \\\\ \hline 0 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 2 & 3 \\\\ 2 & 0 & 2 & 0 & 2 \\\\ 3 & 0 & 3 & 2 & 1 \\\\ \end{array} \] This table shows the results of multiplication under modulo \(4\).