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# If x is an integer and |1-x|<2 then which of the following

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If x is an integer and |1-x|<2 then which of the following [#permalink]  03 Sep 2012, 22:59
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If x is an integer and |1-x|<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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 04 Sep 2012, 02:03, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If x is an integer and |1−x|<2 [#permalink]  04 Sep 2012, 01:11
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sanjoo wrote:
If x is an integer and |1−x|<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|<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.

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Re: If x is an integer and |1−x|<2 [#permalink]  04 Sep 2012, 02:02
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sanjoo wrote:
If x is an integer and |1−x|<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

If x is an integer and |1-x|<2 then which of the following must be true?

|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 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 zeo is a multiple of every integer except zero itself.

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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  15 May 2015, 10:44
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avgroh wrote:
Bunuel wrote:
sanjoo wrote:
If x is an integer and |1−x|<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

If x is an integer and |1-x|<2 then which of the following must be true?

|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 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 zeo is a multiple of every integer except zero itself.

Hi again Bunuel,

Just one question. I know by trial and error that the below process is wrong. But why does the algebra not match the intuitive way of solving??? Could you pls point out where you think I am making an error? TIA.

Given |1-x| < 2

(a) If x>0: 1-x < 2 -> x > -1

But this is true only for x>=0 which is a more limiting condition than x > -1. So shouldn't the result of opening the modulus be x>=0?

(b) If x<0: -1+x < 2 -> x<3

But this is true only for x<0 which is a more limiting condition that x<3. So shouldn't the result of opening the modulus be x<0?

By the above logic x = 0. But I can clearly see that x = 1 and x =2 will also work - why the discrepancy

Hi avgroh,

You are not getting the right result because you are considering the zero points of x when deciding the sign of the expression when it comes out of the modulus. We consider the zero points of the expression inside the modulus to decide the sign of the expression when it comes out of the modulus i.e. in this case zero points of (1-x). So we can solve this modulus as

1. When 1- x > 0
1 - x < 2 i.e. x > -1

2. When 1 - x < 0
-(1-x) < 2 i.e. x < 3

Combining the above we get the range as -1 < x < 3. Since x is an integer it can take values of only 0,1 and 2.

Just to add on to the concept, we write |x| = x if x > 0 and -x if x < 0. On the same line, we would write |x -1| = x-1 if x-1 > 0 and -(x-1) if x-1 < 0

Hope it helps

Regards
Harsh
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Re: If x is an integer and |1-x|<2 then which of the following m [#permalink]  09 Jul 2013, 11:17
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mohitvarshney wrote:

My analysis:
-1<x<3
Possible values of x -> 0, 1 & 2
" E. x is not a multiple of an odd prime number"

x=0 - False as 0 is a multiple of any odd prime number
x=1 - True as 1 is NOT a multiple of any odd prime number
x=2 - True as 2 is not a multiple of any odd prime number.
smallest odd prime number is "3".
So, when x=2 the statement is true.

Is there any flaw in my reasoning?

Apart from the highlighted part, everything is correct. For the reason that you could demonstrate that option E is not valid for x=0, we can not have E as a correct answer for a Must be True type question.

1 is a factor of every integer, not a multiple of every integer.
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  21 Aug 2013, 02:57
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PiyushK wrote:
Hi Bunuel,

I have a doubt, as per this thread D is the correct answer.
I calculated the range of x as -1<x<3, further option D says :
D. Number of distinct positive factors of x+2 is a prime number
for x=0 x+2 = 2 => distinct positive factors are 1 and 2
for x=1 x+2 = 3 => distinct positive factors are 1 and 3
for x=2 x+2 = 4 => distinct positive factors are 1 and 2

2,3 are prime numbers but 1 is not a prime number as per rule/definition.

Therefor I think D is also not a well articulated option.

Regards,
PiyushK

Subject: If x is an integer and |1-x|<2 then which of the following

Bunuel wrote:
sanjoo wrote:
If x is an integer and |1−x|<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

If x is an integer and |1-x|<2 then which of the following must be true?

|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 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 zeo is a multiple of every integer except zero itself.

Responding to pm.

D. Number of distinct positive factors of x+2 is a prime number. x+2 is 2, 3, or 4.

2 has 2 factors 1 and 2.
3 has 2 factors 1 and 3.
4 has 3 factors 1, 2 and 4.

The number of factors of each number is a prime number.

Hope it's clear.
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  22 May 2014, 02:58
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dansa wrote:
E ist also correct1 The question is flawed!!

number 1 is not considered prime, as it has only one factor (itself).

Yes, 1 is NOT prime but it has nothing to do with option E.

E says: x is not a multiple of an odd prime number. IF x=0, then this option is not always true because 0 is a multiple of every integer except 0 itself, hence it's a multiple of all odd primes: 3, 5, 7, ....
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Re: If x is an integer and |1−x|<2 [#permalink]  10 Jun 2014, 05:56
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This might be a naive question and also highlights a gap in my understand but can you please explain how |1−x|<2 translates into "-2<(1-x)<2". Thank you.

SOURH7WK wrote:
sanjoo wrote:
If x is an integer and |1−x|<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|<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.

|1 − x| < 2:

(1 - x) < 2;
-(1 - x) < 2 --> -2 < 1 - x

Hence -2 < 1 - x < 2.

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope this helps.
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Re: If x is an integer and |1−x|<2 [#permalink]  10 Jun 2014, 06:10
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Bunuel - Thank you very much. It absolutely helps (no pun intended); and like I said there is a gap in my understanding since I believed that the absolute value of anything is always positive, hence I was viewing |x-1| as simply (x-1) and did not consider -(x-1). Thanks again.

Absolute value of any number, expression, is more than or equal to zero but the expression in the modulus can be negative as well as positive. So, $$|x-1|\geq{0}$$ but x-1 can be positive negative or 0.
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If x is an integer and |1-x|<2 then which of the following m [#permalink]  09 Jul 2013, 10:21
If x is an integer and |1-x|<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

I am confused between D & E.
D seems perfectly correct.

My analysis:
-1<x<3
Possible values of x -> 0, 1 & 2
" E. x is not a multiple of an odd prime number"

x=0 - False as 0 is a multiple of any odd prime number
x=1 - False as 1 is a multiple of any odd prime number
x=2 - True as 2 is not a multiple of any odd prime number.

smallest odd prime number is "3".
So, when x=2 the statement is true.

Is there any flaw in my reasoning?
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Re: If x is an integer and |1-x|<2 then which of the following m [#permalink]  09 Jul 2013, 12:38
mau5 wrote:
mohitvarshney wrote:

My analysis:
-1<x<3
Possible values of x -> 0, 1 & 2
" E. x is not a multiple of an odd prime number"

x=0 - False as 0 is a multiple of any odd prime number
x=1 - True as 1 is NOT a multiple of any odd prime number
x=2 - True as 2 is not a multiple of any odd prime number.
smallest odd prime number is "3".
So, when x=2 the statement is true.

Is there any flaw in my reasoning?

Apart from the highlighted part, everything is correct. For the reason that you could demonstrate that option E is not valid for x=0, we can not have E as a correct answer for a Must be True type question.

1 is a factor of every integer, not a multiple of every integer.

Yup I got my mistake. It is a "Must be true" question.
Thanks a lot.
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  14 Apr 2014, 03:06
Bunuel,

Why does 2 have 2 factors? I thought the smallest factor of any number was 2.. Is there a definition difference that I am missing?

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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  14 Apr 2014, 03:31
Expert's post
demandi wrote:
Bunuel,

Why does 2 have 2 factors? I thought the smallest factor of any number was 2.. Is there a definition difference that I am missing?

A factor is a positive integer which divides some integer without a remainder. Thus the smallest factor of any integer is 1.

The factors of 2 are 1 and 2.
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  22 May 2014, 02:53
I think E ist also correct. as number 1 is not considered prime, as it has only one factor (itself).

Except if 0 is a multiple of every number!

Last edited by dansa on 22 May 2014, 02:58, edited 1 time in total.
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  22 May 2014, 03:00
I just understood that 0 is considered a multiple of every number.
Thanks
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Re: If x is an integer and |1−x|<2 [#permalink]  09 Jun 2014, 13:10
This might be a naive question and also highlights a gap in my understand but can you please explain how |1−x|<2 translates into "-2<(1-x)<2". Thank you.

SOURH7WK wrote:
sanjoo wrote:
If x is an integer and |1−x|<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|<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.

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Re: If x is an integer and |1−x|<2 [#permalink]  10 Jun 2014, 06:07
Bunuel - Thank you very much. It absolutely helps (no pun intended); and like I said there is a gap in my understanding since I believed that the absolute value of anything is always positive, hence I was viewing |x-1| as simply (x-1) and did not consider -(x-1). Thanks again.
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  11 Jun 2014, 02:16
Bunuel wrote:
dansa wrote:
E ist also correct1 The question is flawed!!

number 1 is not considered prime, as it has only one factor (itself).

Yes, 1 is NOT prime but it has nothing to do with option E.

E says: x is not a multiple of an odd prime number. IF x=0, then this option is not always true because 0 is a multiple of every integer except 0 itself, hence it's a multiple of all odd primes: 3, 5, 7, ....

Bunnel, where do the multiples start for an integer? Say for 3
Do they start at 0 or should the negative multiples be considered too?
0 being the multilple of every integer is certainly a revelation to me. Thanks for that!
Btw, zero has no multiples then?
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  11 Jun 2014, 02:29
Expert's post
Kconfused wrote:
Bunuel wrote:
dansa wrote:
E ist also correct1 The question is flawed!!

number 1 is not considered prime, as it has only one factor (itself).

Yes, 1 is NOT prime but it has nothing to do with option E.

E says: x is not a multiple of an odd prime number. IF x=0, then this option is not always true because 0 is a multiple of every integer except 0 itself, hence it's a multiple of all odd primes: 3, 5, 7, ....

Bunnel, where do the multiples start for an integer? Say for 3
Do they start at 0 or should the negative multiples be considered too?
0 being the multilple of every integer is certainly a revelation to me. Thanks for that!
Btw, zero has no multiples then?

Yes, no number is a multiple of 0.

As for negative multiples: multiples of 3 are: ..., -6, -3, 0, 3, 6, ... But you should not worry about it since every GMAT divisibility question will tell you in advance that any unknowns represent positive integers, which means that ALL GMAT divisibility questions are limited to positive integers only.
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Re: If x is an integer and |1-x|<2 then which of the following [#permalink]  15 May 2015, 08:52
Bunuel wrote:
sanjoo wrote:
If x is an integer and |1−x|<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

If x is an integer and |1-x|<2 then which of the following must be true?

|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 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 zeo is a multiple of every integer except zero itself.

Hi again Bunuel,

Just one question. I know by trial and error that the below process is wrong. But why does the algebra not match the intuitive way of solving??? Could you pls point out where you think I am making an error? TIA.

Given |1-x| < 2

(a) If x>0: 1-x < 2 -> x > -1

But this is true only for x>=0 which is a more limiting condition than x > -1. So shouldn't the result of opening the modulus be x>=0?

(b) If x<0: -1+x < 2 -> x<3

But this is true only for x<0 which is a more limiting condition that x<3. So shouldn't the result of opening the modulus be x<0?

By the above logic x = 0. But I can clearly see that x = 1 and x =2 will also work - why the discrepancy
Re: If x is an integer and |1-x|<2 then which of the following   [#permalink] 15 May 2015, 08:52
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