Question: In a modular system, if a ≡ b (mod n) and c ≡ d (mod n), which of the following is true?
Options:
a + c ≡ b + d (mod n)
a - c ≡ b - d (mod n)
a * c ≡ b * d (mod n)
All of the above
Correct Answer: All of the above
Solution:
All operations maintain the equivalence in modular arithmetic.
In a modular system, if a ≡ b (mod n) and c ≡ d (mod n), which of the following
Practice Questions
Q1
In a modular system, if a ≡ b (mod n) and c ≡ d (mod n), which of the following is true?
a + c ≡ b + d (mod n)
a - c ≡ b - d (mod n)
a * c ≡ b * d (mod n)
All of the above
Questions & Step-by-Step Solutions
In a modular system, if a ≡ b (mod n) and c ≡ d (mod n), which of the following is true?
Step 1: Understand what 'a ≡ b (mod n)' means. It means that when you divide a and b by n, they leave the same remainder.
Step 2: Understand what 'c ≡ d (mod n)' means. It means that when you divide c and d by n, they also leave the same remainder.
Step 3: Know that in modular arithmetic, you can perform operations like addition, subtraction, and multiplication on both sides of the equivalence without changing the truth of the statement.
Step 4: For example, if you add a and c, and b and d, you can say 'a + c ≡ b + d (mod n)'.
Step 5: Similarly, you can subtract or multiply the numbers and the equivalence will still hold true.
Step 6: Therefore, all operations (addition, subtraction, multiplication) maintain the 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 equivalence relation in modular arithmetic allows for the manipulation of congruences, meaning if two numbers are congruent modulo n, their sums, differences, and products are also congruent modulo n.
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