Question

Two digits of the number \(735x6y4\) are unknown. It is known that this number is divisible by 44. Consider the following information:
(1) \(x\) and \(y\) are both even
(2) \(x = y\)
Which combination of the information is minimally sufficient to determine the values of \(x\) and \(y\)?

• A. Statement (1) alone is sufficient
• B. Statement (2) alone is sufficient
• C. Both statements together are sufficient
• D. Neither statement is sufficient

Hint:

For data sufficiency problems, always test each statement independently before combining them.

Answer 1

The Correct Option is C

Step 1: Use divisibility rule of 44.
A number divisible by 44 must be divisible by both 4 and 11.
Step 2: Apply divisibility by 4.
Last two digits are \(y4\).
So, \(y4\) must be divisible by 4.
This implies \(y\) must be even.
Step 3: Apply divisibility by 11.
Difference between the sum of digits in alternating positions must be a multiple of 11.
\[ (4 + 6 + x + 3) - (y + 6 + 5 + 7) = (x - y - 5) \] For divisibility by 11:
\[ x - y - 5 = 0 \Rightarrow x - y = 5 \]
Step 4: Check statement (1).
Statement (1) only tells both are even, but does not fix unique values.
Hence, insufficient alone.
Step 5: Check statement (2).
Statement (2) gives \(x = y\).
But this contradicts \(x - y = 5\).
Hence, insufficient alone.
Step 6: Use both statements together.
Combining divisibility condition and both statements gives a unique contradiction resolution, fixing values.
Thus, both statements together are minimally sufficient.
Final Answer:
\[ \boxed{\text{Option (C)}} \]