In a modular arithmetic system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is not necessarily true?
Practice Questions
1 question
Q1
In a modular arithmetic system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is not necessarily true?
a + c ≡ b + d (mod m)
a - c ≡ b - d (mod m)
a * c ≡ b * d (mod m)
a / c ≡ b / d (mod m)
Division is not guaranteed in modular arithmetic, as it requires the divisor to have a multiplicative inverse.
Questions & Step-by-step Solutions
1 item
Q
Q: In a modular arithmetic system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is not necessarily true?
Solution: Division is not guaranteed in modular arithmetic, as it requires the divisor to have a multiplicative inverse.
Steps: 5
Step 1: Understand what 'a ≡ b (mod m)' means. It means that when you divide (a - b) by m, the remainder is 0.
Step 2: Understand what 'c ≡ d (mod m)' means. It means that when you divide (c - d) by m, the remainder is 0.
Step 3: Recognize that in modular arithmetic, you can add, subtract, and multiply congruences. For example, if a ≡ b and c ≡ d, then (a + c) ≡ (b + d) and (a * c) ≡ (b * d) (mod m) are true.
Step 4: Understand that division in modular arithmetic is different. To divide by a number, that number must have a multiplicative inverse modulo m. This means there must be another number that, when multiplied by the divisor, gives 1 modulo m.
Step 5: Conclude that while addition, subtraction, and multiplication are always valid in modular arithmetic, division is not guaranteed unless the divisor has a multiplicative inverse.
