Congruence in Modular Arithmetic

Key Concepts

• Congruence Definition: When we say (a \equiv b \ (\text{mod} \ n)), it means that (a) and (b) have the same remainder when divided by (n).

  • Note: Remainders can be negative.
  • (n) should be a positive whole number greater than 1.
  • Example: (n = 2, 3, 4, \ldots)

• Important Points:

  • Do not use (n = 1) as it complicates the math.
  • When divided by (n), if (a \equiv b \ (\text{mod} \ n)), (a) and (b) produce the same remainder.

Interpretations of Congruence

1. Remainders Interpretation:

   • (a) and (b) have the same remainder when divided by (n).
   • Example: (10 \equiv 14 \ (\text{mod} \ 4)) because both give remainder 2.

2. Equation Form:

   • (a \equiv b \ (\text{mod} \ n)) can be expressed as (a = kn + b) for some integer (k).
   • This form is useful for converting congruence to equations.
   • Example: (10 \equiv 14 \ (\text{mod} \ 4)) implies (10 = 14 + (-1 \cdot 4)).

3. Multiple Interpretation:

   • From (a \equiv b \ (\text{mod} \ n)), if you subtract (b) from both sides: (a - b = kn).
   • This implies (a - b) is a multiple of (n), i.e., (n) divides (a - b).
   • Notation: (n \mid (a - b)).
   • Example: (4 | (10 - 14)) because (-4 = 4 \cdot (-1)).

Examples

• Example 1: (10 \equiv 14 \ (\text{mod} \ 4))

  • 10 divided by 4 → remainder 2
  • 14 divided by 4 → remainder 2

• Example 2: (10 \equiv -2 \ (\text{mod} \ 4))

  • (-2 + 4 = 2) which is the remainder.
  • This illustrates handling negative remainders.

• Example 3: Converting to equation form

  • (10 = (-1 \cdot 4) + 14)
  • Here, (k = -1).

• Example 4: Multiple of (n)

  • (4 | (10 - 14) = -4)
  • Shows (n) (4) divides (a - b).

Conclusion

• These methods help us understand and solve modular arithmetic problems.
• Useful for converting congruences to equations and for proofs.

If there are any questions, comments, or suggestions, feel free to ask!