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If \(x\) is a positive integer and \((-1)^{(x^2)} = -1\), which of the following could be the remainder when \(x\) is divided by 8?
I. 1
II. 4
III. 7
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
I. 1
II. 4
III. 7
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
05 Jun 2024, 03:31
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Official Solution:
If \(x\) is a positive integer and \((-1)^{(x^2)} = -1\), which of the following could be the remainder when \(x\) is divided by 8?
I. 1
II. 4
III. 7
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
\((-1)^{(x^2)} = -1\) implies that \(x^2\) is odd (if it were even, then \((-1)^{even} = 1\)). \(x^2 = \text{odd}\), on the other hand, means that \(x\) is odd (since \(x\) is an integer). An odd number divided by 8 can only give odd remainders. That's because we'd have \(\text{odd} = 8q + r\), and since \(8q\) is even, \(r\) must be odd for the sum to result in an odd number.
Answer: D
If \(x\) is a positive integer and \((-1)^{(x^2)} = -1\), which of the following could be the remainder when \(x\) is divided by 8?
I. 1
II. 4
III. 7
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
\((-1)^{(x^2)} = -1\) implies that \(x^2\) is odd (if it were even, then \((-1)^{even} = 1\)). \(x^2 = \text{odd}\), on the other hand, means that \(x\) is odd (since \(x\) is an integer). An odd number divided by 8 can only give odd remainders. That's because we'd have \(\text{odd} = 8q + r\), and since \(8q\) is even, \(r\) must be odd for the sum to result in an odd number.
Answer: D
13 Sep 2025, 10:41
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How is 7 a remainder. Can you please give an example of such a number? Question says positive integer for x, so any value (1, 3, 5, 7, 9 when divide by 8 gives remainder as 1 only)
