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For a nonnegative integer n, if the remainder is 1 when 2n is divided by 3, then which of the following must be true?
I. n is greater than zero. ?
II. 3n = (-3)n ?
III. (√2)n is an integer.
II. 3n = (-3)n ?
III. (√2)n is an integer.
- a)I only
- b)II only
- c)II and III only
- d)I, II, and III
Correct answer is option 'C'. Can you explain this answer?
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For a nonnegative integer n, if the remainder is 1 when 2n is divided ...
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For a nonnegative integer n, if the remainder is 1 when 2n is divided ...
I. n is greater than zero:
This statement must be true. Since n is a nonnegative integer, it cannot be negative. Therefore, n must be greater than zero.
II. 3n = (-3)n:
This statement must be false. If we let n = 0, then 3n = 0 and (-3)n = 1. Therefore, 3n is not equal to (-3)n.
III. 2n = (-1)n:
This statement must be true. If the remainder is 1 when 2n is divided by 3, then 2n = 3k + 1 for some integer k. Rearranging this equation, we get 2n - 1 = 3k. Since 2n - 1 is always an odd number, (-1)n must be equal to 1. Therefore, this statement is true.
In conclusion, the only statement that must be true is III.
This statement must be true. Since n is a nonnegative integer, it cannot be negative. Therefore, n must be greater than zero.
II. 3n = (-3)n:
This statement must be false. If we let n = 0, then 3n = 0 and (-3)n = 1. Therefore, 3n is not equal to (-3)n.
III. 2n = (-1)n:
This statement must be true. If the remainder is 1 when 2n is divided by 3, then 2n = 3k + 1 for some integer k. Rearranging this equation, we get 2n - 1 = 3k. Since 2n - 1 is always an odd number, (-1)n must be equal to 1. Therefore, this statement is true.
In conclusion, the only statement that must be true is III.
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