If a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?
Practice Questions
Q1
If a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?
a + c ≡ b + d (mod m)
a - c ≡ b - d (mod m)
a * c ≡ b * d (mod m)
All of the above
Questions & Step-by-Step Solutions
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. This means that when you divide 'a' and 'b' by 'm', they leave the same remainder.
Step 2: Understand what 'c ≡ d (mod m)' means. This means that when you divide 'c' and 'd' by 'm', they also leave the same remainder.
Step 3: Know that if two numbers are congruent modulo 'm', you can add, subtract, or multiply them, and the result will still be congruent modulo 'm'.
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 congruence under modular arithmetic.
Modular Arithmetic – The study of integers and their equivalence under a modulus, where two numbers are considered equivalent if they have the same remainder when divided by a given number.
Properties of Congruences – The properties that allow for the manipulation of congruences, including the preservation of congruence under addition, subtraction, and multiplication.
