Which of the following pairs of numbers are congruent modulo 6?

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Question: Which of the following pairs of numbers are congruent modulo 6?

Options:

Correct Answer: All of the above

Solution:

All pairs have the same remainder when divided by 6: 8 mod 6 = 2, 14 mod 6 = 2; 10 mod 6 = 4, 16 mod 6 = 4; 12 mod 6 = 0, 18 mod 6 = 0.

Which of the following pairs of numbers are congruent modulo 6?

Which of the following pairs of numbers are congruent modulo 6?

Step 1: Understand what 'congruent modulo 6' means. It means two numbers give the same remainder when divided by 6.
Step 2: Take the first pair of numbers and divide each by 6 to find their remainders.
Step 3: For the first pair, 8 divided by 6 gives a remainder of 2 (8 mod 6 = 2).
Step 4: For the same pair, 14 divided by 6 also gives a remainder of 2 (14 mod 6 = 2).
Step 5: Since both numbers in the first pair have the same remainder, they are congruent modulo 6.
Step 6: Repeat steps 2 to 5 for the second pair of numbers (10 and 16).
Step 7: For 10, 10 divided by 6 gives a remainder of 4 (10 mod 6 = 4).
Step 8: For 16, 16 divided by 6 also gives a remainder of 4 (16 mod 6 = 4).
Step 9: Since both numbers in the second pair have the same remainder, they are congruent modulo 6.
Step 10: Repeat steps 2 to 5 for the third pair of numbers (12 and 18).
Step 11: For 12, 12 divided by 6 gives a remainder of 0 (12 mod 6 = 0).
Step 12: For 18, 18 divided by 6 also gives a remainder of 0 (18 mod 6 = 0).
Step 13: Since both numbers in the third pair have the same remainder, they are congruent modulo 6.

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