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# M28-04

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Math Expert
Joined: 02 Sep 2009
Posts: 50613

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16 Sep 2014, 00:28
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Difficulty:

25% (medium)

Question Stats:

69% (00:33) correct 31% (00:53) wrong based on 248 sessions

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What is the sum of integers $$a$$ and $$b$$ ?

(1) $$|a| = -|b|$$

(2) $$|b| = -|a|$$

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Joined: 02 Sep 2009
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16 Sep 2014, 00:28
Official Solution:

Both statements are saying the exact same thing: $$|a|+|b|=0$$, this to be true, both $$a$$ and $$b$$ must equal to zero.

Why is that? Absolute value is always non-negative - $$|\text{some expression}| \geq 0$$, which means that absolute value is either zero or positive. We have that the sum of two absolute values, or the sum of two non-negative values equals to zero: $$\text{non-negative} + \text{non-negative} = 0$$, obviously both must be zero this equation to hold true.

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18 Apr 2017, 17:20
Bunuel wrote:
Official Solution:

Both statements are saying the exact same thing: $$|a|+|b|=0$$, this to be true, both $$a$$ and $$b$$ must equal to zero.

Why is that? Absolute value is always non-negative - $$|\text{some expression}| \geq 0$$, which means that absolute value is either zero or positive. We have that the sum of two absolute values, or the sum of two non-negative values equals to zero: $$\text{non-negative} + \text{non-negative} = 0$$, obviously both must be zero this equation to hold true.

In other words, is this the logic? Re-arrange the statements to be:

(1) |a| + |b| = 0

(2) |b| + |a| = 0

If you add two positive values and they equal zero, then both values have to be zero
Math Expert
Joined: 02 Sep 2009
Posts: 50613

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18 Apr 2017, 20:22
1
joondez wrote:
Bunuel wrote:
Official Solution:

Both statements are saying the exact same thing: $$|a|+|b|=0$$, this to be true, both $$a$$ and $$b$$ must equal to zero.

Why is that? Absolute value is always non-negative - $$|\text{some expression}| \geq 0$$, which means that absolute value is either zero or positive. We have that the sum of two absolute values, or the sum of two non-negative values equals to zero: $$\text{non-negative} + \text{non-negative} = 0$$, obviously both must be zero this equation to hold true.

In other words, is this the logic? Re-arrange the statements to be:

(1) |a| + |b| = 0

(2) |b| + |a| = 0

If you add two positive values and they equal zero, then both values have to be zero

Two non-negative values. The sum of positive numbers cannot be 0.
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Joined: 04 Feb 2014
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10 Jul 2018, 21:19
Bunuel
I understand the arrangement of the equation made above but I was considering the following:
(1) if a>0, a=-b; -b+b=0
BUT if a<0, -a=-b; a=b; and then the sum is a+a=2a (or b+b=2b)
in this case the answer would be not sufficient. Where is the mistake here?
Thank you
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Joined: 02 Sep 2009
Posts: 50613

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10 Jul 2018, 23:05
1
NatalieStarr wrote:
Bunuel
I understand the arrangement of the equation made above but I was considering the following:
(1) if a>0, a=-b; -b+b=0
BUT if a<0, -a=-b; a=b; and then the sum is a+a=2a (or b+b=2b)
in this case the answer would be not sufficient. Where is the mistake here?
Thank you

If a > 0 and b > 0, then |a| = -|b| transforms to a = -b --> a + b = 0. This case is not possible: the sum of two positive values cannot be 0.
If a > 0 and b < 0, then |a| = -|b| transforms to a = -(-b) --> a = b. This case is not possible: positive a (a > 0) cannot equal to negative b (b < 0).
If a < 0 and b < 0, then |a| = -|b| transforms to -a = -(-b) --> a + b = 0. This case is not possible: the sum of two negative values cannot be 0.
If a < 0 and b > 0, then |a| = -|b| transforms to -a = -b --> a = b. This case is not possible: negative a (a < 0) cannot equal to positive b (b > 0).

The only case left is a = b = 0.
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06 Sep 2018, 03:05
But we are told to find sum of a and b not sum of mod a and mod b..
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06 Sep 2018, 03:08
hellogmat123999 wrote:
But we are told to find sum of a and b not sum of mod a and mod b..

From both statements we got that a = b = 0, thus a + b = 0.
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16 Oct 2018, 05:07
Bunuel wrote:
hellogmat123999 wrote:
But we are told to find sum of a and b not sum of mod a and mod b..

From both statements we got that a = b = 0, thus a + b = 0.

But how do we know a or b aren't anything other than 0? How do we not know a and b can't both equal 1 or 2 or any other integer? Is 0 the only number where two different values (a and b) can be the same? I ask because when two letters are used, it typically implies different values, but I genuinely could have just not known any two differently-labeled values could equal zero.
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16 Oct 2018, 06:25
okayfabian15 wrote:
Bunuel wrote:
hellogmat123999 wrote:
But we are told to find sum of a and b not sum of mod a and mod b..

From both statements we got that a = b = 0, thus a + b = 0.

But how do we know a or b aren't anything other than 0? How do we not know a and b can't both equal 1 or 2 or any other integer? Is 0 the only number where two different values (a and b) can be the same? I ask because when two letters are used, it typically implies different values, but I genuinely could have just not known any two differently-labeled values could equal zero.

This is explained here: https://gmatclub.com/forum/m28-184502.html#p1415666

As for your doubt: unless it is explicitly stated otherwise, different variables CAN represent the same number.
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Re: M28-04 &nbs [#permalink] 16 Oct 2018, 06:25
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# M28-04

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