If x is a positive integer and (-1)^(x^2) = -1, which of the following

Question

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

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Bunuel · Accepted Answer

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 = 	ext{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  	ext{odd} = 8q + r , and since  8q  is even,  r  must be odd for the sum to result in an odd number.

Answer: D

Purnank · Answer

Given - x +ve integer, and (-1)^(x^2) = -1 if you observe carefully when this will happen?? 
when x^2 is odd, means x is +ve odd integer.
and when we divide any +ve odd integer with 8 we always get the remainder in the form of odd number.
for eg. 9 divided by 8 remainder is 1, Similarly 7 divided by 8 remainder is 7.
Answer D.

Scholar94 · Answer

Answer: D

What the expression simply mean is that X is an odd integer.

Given that this is a could be true question, we cab test number. Also given the range of options we are given, we can start picking a odd number just above 8.

If x=9, 9/8 gives a remainder of 1. Thus, I. Could be true. If x=11, 11/8, gives a remainder of 3. But 3 is not in the option. 13/8 gives a reminder of 5. This signal that 4 cannot be a remainder. So II can't be true. Also if x=15, 15/8 gives a reminder of 7. Thus III could be true. HenceD is correct

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gmatophobia · Answer

(-1)^{x^2} = -1

Hence, we can conclude that  x^2 =  odd

Odd = 8*quotient + remainder

Hence, remainder = odd - even = odd

Out of the given numbers, 1 and 7 are odd.

Option D.

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If x is a positive integer and (-1)^(x^2) = -1, which of the following

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GMAT Problem Solving (PS) Questions

If x is a positive integer and (-1)^(x^2) = -1, which of the following

Prep Toolkit

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My Rewards