Question

Consider a seven-digit number 735x6y4, divisible by 44, where the two digits \( x \) and \( y \) are unknown.
Consider the following two additional pieces of information:
I. \( x \) and \( y \) are even numbers.
II. \( x \) and \( y \) are equal.
To determine the values of \( x \) and \( y \) UNIQUELY, which of the above pieces of information is/are MINIMALLY SUFFICIENT?

• A. The answer cannot be obtained even with both additional pieces of information
• B. II only
• C. Either I or II, by itself, is minimally sufficient
• D. Both I and II together
• E. I only

Hint:

For divisibility problems, always check divisibility by each factor (in this case 4 and 11) separately and combine the results.

Answer 1

The Correct Option is A

Step 1: Understand divisibility by 44.
A number is divisible by 44 if it is divisible by both 4 and 11. - For divisibility by 4, the last two digits must be divisible by 4. - For divisibility by 11, the alternating sum of the digits must be divisible by 11.
Step 2: Apply the given conditions.
- \( x \) and \( y \) are even, so \( x = 0, 2, 4, 6, \) or \( 8 \) and \( y = 0, 2, 4, 6, \) or \( 8 \). - \( x = y \). Even with both pieces of information, there are still multiple combinations of \( x \) and \( y \) that satisfy the divisibility conditions. Thus, the answer cannot be uniquely determined.
Step 3: Conclusion.
The correct answer is (A).