In a certain modular arithmetic system, if 7 is congruent to 3 modulo n, what ca

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Question: In a certain modular arithmetic system, if 7 is congruent to 3 modulo n, what can be inferred about n?

Options:

Correct Answer: n must be greater than 4

Solution:

Since 7 - 3 = 4, n must be a divisor of 4. Therefore, n can be 1, 2, or 4, but for the congruence to hold, n must be greater than 4.

In a certain modular arithmetic system, if 7 is congruent to 3 modulo n, what can be inferred about n?

In a certain modular arithmetic system, if 7 is congruent to 3 modulo n, what can be inferred about n?

Step 1: Understand what 'congruent' means in modular arithmetic. It means two numbers have the same remainder when divided by a certain number, n.
Step 2: Write down the given information: 7 is congruent to 3 modulo n.
Step 3: This means that when you subtract 3 from 7, the result (which is 4) must be divisible by n.
Step 4: Calculate the difference: 7 - 3 = 4.
Step 5: Since n must divide 4, list the divisors of 4. The divisors are 1, 2, and 4.
Step 6: However, for the congruence to hold, n must be greater than 4. Therefore, n cannot be 1, 2, or 4.
Step 7: Conclude that n must be a number greater than 4.

Modular Arithmetic – The study of integers and their equivalence under a modulus, where two numbers are considered equivalent if their difference is divisible by the modulus.
Divisibility – Understanding that if a number is congruent to another modulo n, the difference between the two numbers must be divisible by n.