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# If x is an integer, what is the remainder when |1 - x^2| is divided by

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If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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15 May 2015, 04:32
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75% (hard)

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54% (02:09) correct 46% (02:28) wrong based on 166 sessions

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If x is an integer, what is the remainder when |1 - x^2| is divided by 4?

(1) The sum of any two factors of x is even
(2) The product of any two factors of x is odd

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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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15 May 2015, 09:48
Bunuel wrote:
If x is an integer, what is the remainder when |1 - x^2| is divided by 4?

(1) The sum of any two factors of x is even
(2) The product of any two factors of x is odd

Kudos for a correct solution.

I'll try:

(1) The sum of any two factors of x is even
Factors of X can be 2 and 2 (4) or 3 and 3 (6). This also results in a non defined result for x^2 (can be odd or even).

(2) The product of any two factors of x is odd
For this to be true, any factor of x has to be odd. This statement is sufficient because x^2 will always be odd. Combine this with the question: |1 - x^2| > what is the remainder? The remainder will always be 0 since 1 - ODD = EVEN.
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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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15 May 2015, 10:11
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Bunuel wrote:
If x is an integer, what is the remainder when |1 - x^2| is divided by 4?

(1) The sum of any two factors of x is even
(2) The product of any two factors of x is odd

Kudos for a correct solution.

We can write this as (1 - x)(1+x)/4 or -[(1-x)(1+x)]/4

A) Sum of two factors of x is even, factors of x includes 1 and x itself, since Odd + Odd = Even => x is odd. (as 1 is a factor, so we can't consider Even + Even = Even)
Now, since x is odd => x+1 and x-1(or 1-x) are both even => it will be divisible by 4 ---- Sufficient

B) Product of two factors of x is even, since Odd * Odd = Odd => x is odd.
Now, since x is odd => x+1 and x-1(or 1-x) are both even => it will be divisible by 4 ---- Sufficient

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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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15 May 2015, 10:36
Reminder=?; so question is x=?

1. Sum of any 2 factors are Even, x could be 2+2,3+3 so not sufficient
2. Product of any 2 factors are Odd, x could be 3*3, 5*3 or odd* odd only;as one even* odd =Even; but still we do not know value of x; Not sufficient

1+2 only value possible is 3+3 or 3*3 or 5+5 or 5*5 so still not sufficient.

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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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15 May 2015, 11:13
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Hi reto & lipsi18,

You have to be very careful about the wording in this prompt.

Fact 1 tells us that the sum of ANY two factors of X is EVEN.

Since the number 1 is a factor of every integer, you have to account for that possibility in your work. In addition, you've both used "duplicate" factors, which is not mathematically correct.

For example, the factors of 6 are 1, 2, 3 and 6.....NOT 1, 2, 2, 3, 3 and 6.

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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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15 May 2015, 21:02
Bunuel wrote:
If x is an integer, what is the remainder when |1 - x^2| is divided by 4?

(1) The sum of any two factors of x is even
(2) The product of any two factors of x is odd

Ans: D

Solution: given equation |1-x^2| = |(1-x)(1+x)|
now we know that one factor for the any given number is always 1. so other factors can be any integer value.
now we work with the options

1) sum of any two is even. this statement can be true only if other factors are odd because one factor known to us '1' is odd and only (odd+odd=even)
if any factor is even then (1+even=odd) goes against the statement (1).
now as we know that other factors are also odd and (1+odd) & (1-odd) always even.
by putting this in the equation |(1-x)(1+x)| we can say that reminder will be zero. multiplication of two even integer is always divisible by 4. [Sufficient]

2) product of any two factors is odd. now we know for product to be always odd, both integers must be odd.
again odd+1 and odd-1 will be even and remainder will be zero again. [Sufficient]

so Ans D [Both statements alone are sufficient to answer the question]
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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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16 May 2015, 09:19

statment 1 means that x is odd and we will find that the result of |1-x^2| always divided by 4.
statment two match with what statment 1 included x is odd number
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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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17 May 2015, 04:23
+1 for D.

Both statements are telling us the same, that no factor 2 is present in x. Therefore, pluggin in easy numbers such 3 or 5 we can see that the remainder will always be 0.
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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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18 May 2015, 08:40
Bunuel wrote:
If x is an integer, what is the remainder when |1 - x^2| is divided by 4?

(1) The sum of any two factors of x is even
(2) The product of any two factors of x is odd

Kudos for a correct solution.

OFFICIAL SOLUTION:

If x is an integer, what is the remainder when |1 - x^2| is divided by 4?

Notice that if x is odd, then |1 - x^2| is a multiple of 4. For example:
If x=1, |1 - x^2| = 0;
If x=3, |1 - x^2| = 8;
If x=5, |1 - x^2| = 24.
...

(1) The sum of any two factors of x is even. For the sum of ANY two factors of x to be even all factors of x must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that x is an odd number. Sufficient.

(2) The product of any two factors of x is odd. Basically the same here: for the product of ANY two factors of x to be odd all factors of x must be odd (even if one of the factors is even then we could pair that even factor with any other factor and we'd get even product), which means that x is an odd number. Sufficient.

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Re: If x is an integer, what is the remainder when |1 - x^2| is divided by  [#permalink]

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27 Sep 2017, 07:57
Bunuel wrote:
If x is an integer, what is the remainder when |1 - x^2| is divided by 4?

(1) The sum of any two factors of x is even
(2) The product of any two factors of x is odd

Kudos for a correct solution.

So it's clear that both statements mean the exact same thing and do no contribute any new information to each other- which reduces our options to either E or D. All we need to know is that X is odd because any odd value plugged in will be a multiple of 4

D
Re: If x is an integer, what is the remainder when |1 - x^2| is divided by &nbs [#permalink] 27 Sep 2017, 07:57
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