Which of the following equations is true in modular arithmetic?

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Question: Which of the following equations is true in modular arithmetic?

Options:

Correct Answer: 8 ≡ 15 (mod 8)

Solution:

8 ≡ 15 (mod 8) is false; the correct answer is 5 ≡ 10 (mod 5) is true.

Which of the following equations is true in modular arithmetic?

Which of the following equations is true in modular arithmetic?

Step 1: Understand what modular arithmetic means. It involves finding the remainder when one number is divided by another.
Step 2: Look at the first equation: 8 ≡ 15 (mod 8). This means we need to check if 15 and 8 give the same remainder when divided by 8.
Step 3: Calculate the remainder of 15 when divided by 8. 15 divided by 8 is 1 with a remainder of 7.
Step 4: Calculate the remainder of 8 when divided by 8. 8 divided by 8 is 1 with a remainder of 0.
Step 5: Compare the remainders. Since 7 (from 15) is not equal to 0 (from 8), the equation 8 ≡ 15 (mod 8) is false.
Step 6: Now look at the second equation: 5 ≡ 10 (mod 5). We need to check if 10 and 5 give the same remainder when divided by 5.
Step 7: Calculate the remainder of 10 when divided by 5. 10 divided by 5 is 2 with a remainder of 0.
Step 8: Calculate the remainder of 5 when divided by 5. 5 divided by 5 is 1 with a remainder of 0.
Step 9: Compare the remainders. Since both 10 and 5 give a remainder of 0, the equation 5 ≡ 10 (mod 5) is true.

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