If a ≡ b (mod m) and c ≡ d (mod m), which of the following is necessarily true?
Practice Questions
Q1
If a ≡ b (mod m) and c ≡ d (mod m), which of the following is necessarily 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 necessarily 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 congruent to each other, you can add, subtract, or multiply them, and the result will still be congruent to the same modulus.
Step 4: For addition, if a ≡ b (mod m) and c ≡ d (mod m), then (a + c) ≡ (b + d) (mod m) is true.
Step 5: For subtraction, if a ≡ b (mod m) and c ≡ d (mod m), then (a - c) ≡ (b - d) (mod m) is true.
Step 6: For multiplication, if a ≡ b (mod m) and c ≡ d (mod m), then (a * c) ≡ (b * d) (mod m) is true.
Step 7: Therefore, all operations (addition, subtraction, multiplication) preserve congruence in 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 rules that govern how congruences behave under various operations such as addition, subtraction, and multiplication.
