# Math Multiple Choice

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 4^{1+1} = 4^{2} = 16. So I is correct. The value of *y* could be 3 if *x* = 2, since 2^{3+1} = 2^{4} = 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)