Question

Directions: Each of the following questions is followed by two statements, labelled (A) and (B). Decide whether the statements are sufficient to conclusively answer the question, and choose:
(A) if Statement (A) alone is sufficient but Statement (B) alone is not.
(B) if Statement (B) alone is sufficient but Statement (A) alone is not.
(C) if Statement (A) and Statement (B) together are sufficient, but neither alone is sufficient.
(D) if either Statement (A) alone or Statement (B) alone is sufficient.
(E) if both statements together are still not sufficient.

Let \(f : N \to N\) (N is the set of all natural numbers). How many solutions are there to the equation \(f(x) = 1485\)?
(A) For \(a, b \in N\), \(f(10a+b) = f(a) + 12b\).
(B) The maximum value of b is 9.

• (A) Statement (A) alone is sufficient, but Statement (B) alone is not sufficient.
• (B) Statement (B) alone is sufficient, but Statement (A) alone is not sufficient.
• (C) Statement (A) and Statement (B) together are sufficient, but neither alone is sufficient.
• (E) Both Statement (A) and Statement (B) together are not sufficient to answer the question.

Hint:

Try unwinding the recursion f(10a+b) = f(a) + 12b for x = 1485 digit by digit and see what value you eventually need to already know.

Answer 1

The Correct Option is D

Step 1: Understand the question.
We want to count how many natural numbers x satisfy \(f(x) = 1485\). We are given a possible recursive rule linking f at a bigger number to f at a smaller one.

Step 2: Test Statement A alone.
Statement A says that if we write \(x = 10a+b\) (splitting off the last digit b, with a being the number formed by the remaining digits), then \(f(x) = f(a) + 12b\). This is a recursive rule: to know f(x), we need f(a), and to know f(a) we need f of a's leading digits, and so on, eventually needing to know f at the one-digit starting values. But we are never told what f(0), f(1), ..., f(9) actually are. Without this base case, the recursion cannot be unwound to an actual number, so we cannot compute f(x) or solve \(f(x)=1485\). Not sufficient alone.

Step 3: Test Statement B alone.
Statement B just tells us b, a digit, goes up to 9 at most (which is true of any digit anyway), and gives no formula connecting f to x at all. Not sufficient alone.

Step 4: Test both statements together.
Even combining both, we still only know the recursive step \(f(10a+b) = f(a) + 12b\) and that \(b \le 9\); we are never given a starting value for f on the single digits, so the recursion can never be fully unwound into an actual numeric formula for f(x). This means f(x) = 1485 cannot actually be solved, and worse, we cannot even determine how many solutions there would be, since the missing base case leaves f itself undetermined.

Final Answer:
\[ \boxed{\text{Both statements together are not sufficient}} \]