In modular arithmetic, which of the following is true for any integer a?

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Question: In modular arithmetic, which of the following is true for any integer a?

Options:

Correct Answer: a mod 2 = 0 or 1

Solution:

For any integer a, a mod 2 will always yield either 0 or 1, depending on whether a is even or odd.

In modular arithmetic, which of the following is true for any integer a?

In modular arithmetic, which of the following is true for any integer a?

Step 1: Understand what 'mod' means. 'a mod b' means the remainder when a is divided by b.
Step 2: Identify the value of b in this case. Here, b is 2.
Step 3: Recognize that any integer can be either even or odd.
Step 4: If a is even, when you divide it by 2, the remainder is 0. So, a mod 2 = 0.
Step 5: If a is odd, when you divide it by 2, the remainder is 1. So, a mod 2 = 1.
Step 6: Conclude that for any integer a, a mod 2 will always be either 0 (if a is even) or 1 (if a is odd).

Modular Arithmetic – A system of arithmetic for integers, where numbers wrap around after reaching a certain value (the modulus).
Even and Odd Numbers – Even numbers are divisible by 2, yielding a remainder of 0 when divided by 2, while odd numbers yield a remainder of 1.