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?
C
Step-by-Step Solution
Key idea: This is a divisibility and data sufficiency problem. The trigger is a number with unknown digits and conditions to uniquely determine them.
Step 1: Analyze the base divisibility condition.
The number is divisible by .
Divisibility by 4: The last two digits must be divisible by 4. This means is a multiple of 4, so must be an even digit ().
Divisibility by 11: The alternating sum of digits must be a multiple of 11.
Odd positions: .
Even positions: .
Difference: .
For this to be a multiple of 11, and knowing , the only possible value is .
Step 2: Evaluate Statement I.
Statement I says and are even.
From , if is even, is also even. Thus, all valid pairs already satisfy this condition. Statement I provides no new information and cannot uniquely determine and .
Step 3: Evaluate Statement II.
Statement II says .
Substituting into gives . This uniquely determines the digits.
Step 4: Conclusion.
Statement II alone is minimally sufficient.
Answer: II only (Option C).