# SAT Math Multiple Choice Question 642: Answer and Explanation

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**Question: 642**

**12.** If *i*^{2}^{k} = 1, and , which of the following must be true about *k*?

- A.
*k*is a multiple of 4. - B.
*k*is a positive integer. - C. When 2
*k*is divided by 4, the remainder is 1. - D. is an integer.

**Correct Answer:** D

**Explanation:**

**D**

**Special Topics (complex numbers) MEDIUM-HARD**

Recall from Chapter 10, Lesson 10, that *i*^{n} = 1 if and only if *n* is a multiple of 4. (If you need refreshing, just confirm that *i*^{4} = 1, *i*^{8} = 1, *i*^{12} = 1, etc.) Therefore, if *i*^{2k} = 1, then *2k* must be a multiple of 4, and therefore, *k* must be a multiple of 2. If *k* is a multiple of 2, then *k*/2 must be an integer. Choice (A) is incorrect, because *k* = 2 is a solution, but 2 is not a multiple of 4. Choice (B) is incorrect because *k* = -2 is a solution, and -2 is not a positive integer. Choice (C) is incorrect because *k* = 2 is a solution, but when 2(2) = 4 is divided by 4, the remainder is 0, not 1.