If a ≡ 2 (mod 3) and b ≡ 1 (mod 3), what is the value of (a * b) mod 3?

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Question: If a ≡ 2 (mod 3) and b ≡ 1 (mod 3), what is the value of (a * b) mod 3?

Options:

Correct Answer: 1

Solution:

a * b ≡ 2 * 1 ≡ 2 (mod 3).

If a ≡ 2 (mod 3) and b ≡ 1 (mod 3), what is the value of (a * b) mod 3?

If a ≡ 2 (mod 3) and b ≡ 1 (mod 3), what is the value of (a * b) mod 3?

Step 1: Understand what 'a ≡ 2 (mod 3)' means. This means when you divide 'a' by 3, the remainder is 2.
Step 2: Understand what 'b ≡ 1 (mod 3)' means. This means when you divide 'b' by 3, the remainder is 1.
Step 3: Multiply the values of 'a' and 'b' that we have in terms of their remainders: 2 (from 'a') and 1 (from 'b').
Step 4: Calculate the product: 2 * 1 = 2.
Step 5: Now, we need to find (a * b) mod 3. Since we calculated a * b = 2, we find 2 mod 3.
Step 6: To find 2 mod 3, we see how many times 3 fits into 2. It fits 0 times, and the remainder is 2.
Step 7: Therefore, (a * b) mod 3 = 2.

Modular Arithmetic – Understanding how to perform operations under a modulus, specifically how to multiply and reduce results.