In a modular system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following

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Question: In a modular system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?

Options:

Correct Answer: All of the above

Solution:

All operations (addition, subtraction, multiplication) maintain equivalence in modular arithmetic.

In a modular system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?

In a modular system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?

Step 1: Understand what 'a ≡ b (mod m)' means. It means that when you divide 'a' and 'b' by 'm', they leave the same remainder.
Step 2: Understand what 'c ≡ d (mod m)' means. It means that when you divide 'c' and 'd' by 'm', they also leave the same remainder.
Step 3: Know that in modular arithmetic, if two numbers are equivalent (like a and b), you can perform operations on them and the equivalence will still hold.
Step 4: For addition: If a ≡ b (mod m) and c ≡ d (mod m), then (a + c) ≡ (b + d) (mod m).
Step 5: For subtraction: If a ≡ b (mod m) and c ≡ d (mod m), then (a - c) ≡ (b - d) (mod m).
Step 6: For multiplication: If a ≡ b (mod m) and c ≡ d (mod m), then (a * c) ≡ (b * d) (mod m).
Step 7: Therefore, all operations (addition, subtraction, multiplication) maintain equivalence in modular arithmetic.

Modular Arithmetic – Modular arithmetic involves integers and a modulus, where two numbers are considered equivalent if they have the same remainder when divided by the modulus.
Equivalence Relations – The properties of equivalence relations, such as reflexivity, symmetry, and transitivity, are fundamental in understanding how equivalence classes work in modular systems.
Operations in Modular Arithmetic – Addition, subtraction, and multiplication of congruences preserve equivalence, meaning if a ≡ b (mod m) and c ≡ d (mod m), then operations on these congruences yield results that are also congruent modulo m.