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The first four digits of the six-digit initial password for a shopper's card at a certain grocery store is the customer's birthday in day-month digit form. For example, 15 August corresponds to 1508 and 5 March corresponds to 0503. The 5th digit of the initial password is the units digit of seven times the sum of the first and third digits, and the 6th digit of the initial password is the units digit of three times the sum of the second and fourth digits. What month, and what day of that month, was a customer born whose initial password ends in 16?
We need to find a customer's birth date (month and day) given that their password ends in "16".
We need to find the unique values for DD (day) and MM (month).
For sufficiency: We need exactly one valid date that produces a password ending in "16".
Since the password ends in "16":
These constraints significantly limit our possible dates.
Statement 1 tells us: The password begins with "21" and its fourth digit is 1.
This gives us the partial date: 21_1
Since the 5th digit must be 1, we need (1st digit + 3rd digit) to equal either 3 or 13.
Since digits must be single values (0-9), the 3rd digit must equal 1.
This gives us the complete date: 21/01 (January 21st)
Let's verify this produces a password ending in 16:
Statement 1 gives us exactly one valid date.
[STOP - Statement 1 is Sufficient!]
This eliminates choices B, C, and E.
Important: We now forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: The sum of the first and third digits is 3, and the second digit is 1.
This gives us:
We need to determine which combination of 1st and 3rd digits (that sum to 3) creates a valid date.
With 2nd digit = 1, possible days are: 01, 11, 21, 31
For (1st + 3rd) = 3, let's check each possibility:
We have three valid dates that satisfy Statement 2's conditions. Since we cannot determine a unique date, Statement 2 alone is NOT sufficient.
This eliminates choices B and D.
Statement 1 alone gives us exactly one date (January 21st), while Statement 2 alone gives us three possible dates.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."