CollegeBoard

For positive integers a and b, let a  b be defined as a^{b +1}. If x and y are positive integers and x  y = 16, which of the following could be a value of y?

I.	1
II.	2
III.	3

• (A) I only
• (B) II only
• (C) I and III only    CORRECT ANSWER
• (D) II and III only
• (E) I, II, and III

Explanation:

For this question, you are given that x  y = 16 where x  y is defined as x^{y +1}. You are asked which of three values are possible for y when x^{y +1} = 16.

The value of y could be 1 if x = 4, since 4¹⁺¹ = 4² = 16. So I is correct. The value of y could be 3 if x = 2, since 2³⁺¹ = 2⁴ = 16. So III is correct. Since there is no integer that can be raised to the (2 + 1) or 3rd power to obtain 16, II is not correct. The correct answer is (C).

(from the October 16, 1999 test)