Question

Let \( p = 4 \times 6 \times 11 \times n \), where \( n \) is a positive integer. Compare the following:
Quantity A: Remainder when \( p \) is divided by 5
Quantity B: Remainder when \( p \) is divided by 33

• A. Quantity A is greater.
• B. Quantity B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined.

Hint:

When comparing remainders, check divisibility carefully. Fixed divisibility often gives consistent results.

Answer 1

The Correct Option is B

Step 1: Simplify \( p = 4 \times 6 \times 11 \times n = 264n \).

Step 2: Find remainder when divided by 5. \( 264n \div 5 \). Since \( 264 \equiv 4 \pmod{5} \), remainder = \( 4n \pmod{5} \), varies with \( n \). Could be 0,1,2,3,4. 

Step 3: Divide by 33. Since \( 264n = 33 \times 8n \), always divisible by 33, remainder = 0. 

Step 4: Compare: Quantity B = 0 always. Quantity A varies but is nonnegative and can be \(>0 \). So Quantity \(A ≥0 \), Quantity B =0. But for \( n \) not multiple of 5, \(A>B \). 
Final Answer: \[ \boxed{\text{Quantity A is greater (except when \( n \) multiple of 5, then equal)}} \]