Question

\( n \) is a positive integer that is divisible by 6. Compare:
Quantity A: The remainder when \( n \) is divided by 12
Quantity B: The remainder when \( n \) is divided by 18

• A. Quantity A is greater.
• B. Quantity B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.
• E.

Hint:

When dealing with remainders, always test multiple values consistent with the given conditions. Often, different values give different outcomes, making the relationship indeterminate.

Answer 1

The Correct Option is D

Step 1: Condition.
It is given that \( n \) is a positive integer divisible by 6. Hence possible values are: \( 6, 12, 18, 24, 30, 36, \dots \).
Step 2: Remainder when divided by 12.
Dividing such numbers by 12, the possible remainders are either 0 or 6. Thus Quantity A can be 0 or 6.
Step 3: Remainder when divided by 18.
Dividing such numbers by 18, the possible remainders are 0, 6, or 12. Thus Quantity B can be 0, 6, or 12.
Step 4: Compare values.
- If \( n = 36 \), remainders are both 0 → Quantities equal.
- If \( n = 18 \), remainders are 6 and 0 → Quantity A>Quantity B.
- If \( n = 30 \), remainders are 6 and 12 → Quantity B>Quantity A.
Step 5: Conclusion.
Since all three cases (A greater, B greater, equal) are possible, the relationship cannot be determined. \[ \boxed{\text{(D) The relationship cannot be determined.}} \]