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

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Question: In a modular arithmetic system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is not necessarily true?

Options:

Correct Answer: a / c ≡ b / d (mod m)

Solution:

Division is not guaranteed in modular arithmetic, as it requires the divisor to have a multiplicative inverse.

In a modular arithmetic system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is not necessarily true?

In a modular arithmetic system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is not necessarily true?

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.

Modular Arithmetic – A system of arithmetic for integers, where numbers wrap around upon reaching a certain value (the modulus).
Congruence Relations – The relationship between two numbers that have the same remainder when divided by a modulus.
Multiplicative Inverses – An element 'x' is a multiplicative inverse of 'a' modulo 'm' if (a * x) ≡ 1 (mod m).