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If \(x\) is an integer and \(|1-x| \lt 2\), then which of the following must be true?
A. \(x\) is not a prime number
B. \(x^2+x\) is not a prime number
C. \(x\) is positive
D. Number of distinct positive factors of \(x+2\) is a prime number
E. \(x\) is not a multiple of an odd prime number
A. \(x\) is not a prime number
B. \(x^2+x\) is not a prime number
C. \(x\) is positive
D. Number of distinct positive factors of \(x+2\) is a prime number
E. \(x\) is not a multiple of an odd prime number
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04 Sep 2012, 03:02
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Official Solution:
If \(x\) is an integer and \(|1-x| \lt 2\), then which of the following must be true?
A. \(x\) is not a prime number
B. \(x^2+x\) is not a prime number
C. \(x\) is positive
D. Number of distinct positive factors of \(x+2\) is a prime number
E. \(x\) is not a multiple of an odd prime number
\(|1-x|\) is just the distance between 1 and \(x\) on the number line. We are told that this distance is less than 2:
----(-1)------1------3----
So, \(-1 < x < 3\). Since it is given that \(x\) is an integer, then \(x\) can be 0, 1 or 2.
A. \(x\) is not a prime number. Not true if \(x=2\).
B. \(x^2+x\) is not a prime number. Not true if \(x=1\).
C. \(x\) is positive. Not true if \(x=0\).
D. Number of distinct positive factors of \(x+2\) is a prime number. True for all three values of \(x\).
E. \(x\) is not a multiple of an odd prime number. Not true if \(x=0\), since zero is a multiple of every integer.
Answer: D
If \(x\) is an integer and \(|1-x| \lt 2\), then which of the following must be true?
A. \(x\) is not a prime number
B. \(x^2+x\) is not a prime number
C. \(x\) is positive
D. Number of distinct positive factors of \(x+2\) is a prime number
E. \(x\) is not a multiple of an odd prime number
\(|1-x|\) is just the distance between 1 and \(x\) on the number line. We are told that this distance is less than 2:
----(-1)------1------3----
So, \(-1 < x < 3\). Since it is given that \(x\) is an integer, then \(x\) can be 0, 1 or 2.
A. \(x\) is not a prime number. Not true if \(x=2\).
B. \(x^2+x\) is not a prime number. Not true if \(x=1\).
C. \(x\) is positive. Not true if \(x=0\).
D. Number of distinct positive factors of \(x+2\) is a prime number. True for all three values of \(x\).
E. \(x\) is not a multiple of an odd prime number. Not true if \(x=0\), since zero is a multiple of every integer.
Answer: D
04 Sep 2012, 02:11
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sanjoo
|1−x|<2 = -2<(1-x)<2
= -3<-x<1
= 3>x>-1
So x can hold values of 0,1 & 2 to satisfy the condition. Now we can evaluate the choices.
A) 1 & 2 primes, so incorrect
B) 1^2+1=2 is a prime, so incorrect
C) 0 is not +ve, So incorrect
D) x+2= 2,3,or 4, here 2 has 2 factor(prime), 3 has 2 factor (prime) & 4 has 3factors (prime). Hence correct choice.
E) 2 is multiple of 1. So incorrect.
Hence Answer D.
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