When every even power of every odd integer (greater than 1) is divided by 8, what is the remainder?
- A. 3
- B. 2
- C. 1 ✓
- D. The remainder is not necessarily 1.
Correct Answer: C. 1
Explanation
An odd integer can be written as 2k+1. Its square is 4k^2+4k+1 = 4k(k+1)+1. Since k(k+1) is always an even number, 4k(k+1) is a multiple of 8. Thus, (2k+1)^2 \equiv 1 \pmod 8, and any further power also leaves a remainder of 1.
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