In a certain modular arithmetic system, if 7 is congruent to 3 modulo 4, which o

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Question: In a certain modular arithmetic system, if 7 is congruent to 3 modulo 4, which of the following statements is true?

Options:

Correct Answer: 7 - 3 is divisible by 4

Solution:

In modular arithmetic, if a ≡ b (mod m), then (a - b) is divisible by m. Here, 7 - 3 = 4, which is divisible by 4.

In a certain modular arithmetic system, if 7 is congruent to 3 modulo 4, which of the following statements is true?

In a certain modular arithmetic system, if 7 is congruent to 3 modulo 4, which of the following statements is true?

Step 1: Understand what 'congruent' means in modular arithmetic. It means that two numbers have the same remainder when divided by a certain number (the modulus).
Step 2: Identify the numbers in the question. We have 7 and 3, and the modulus is 4.
Step 3: Calculate the difference between the two numbers: 7 - 3 = 4.
Step 4: Check if this difference (4) is divisible by the modulus (4). Since 4 divided by 4 equals 1, it is divisible.
Step 5: Conclude that since 7 - 3 is divisible by 4, the statement '7 is congruent to 3 modulo 4' is true.

Modular Arithmetic – A system of arithmetic for integers, where numbers wrap around upon reaching a certain value (the modulus).
Congruence Relation – A relation that indicates two numbers have the same remainder when divided by a modulus.
Divisibility – A property indicating that one integer can be divided by another without leaving a remainder.