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Math Expert V
Joined: 02 Sep 2009
Posts: 59126

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Difficulty:   25% (medium)

Question Stats: 66% (00:53) correct 34% (01:15) wrong based on 178 sessions

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

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

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

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Math Expert V
Joined: 02 Sep 2009
Posts: 59126

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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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Math Expert V
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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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Math Expert V
Joined: 02 Sep 2009
Posts: 59126

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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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Joined: 01 Nov 2016
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Concentration: Technology, Operations

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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
Intern  B
Joined: 04 Feb 2014
Posts: 2
Location: Spain

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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
Intern  B
Joined: 17 Apr 2018
Posts: 2

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But we are told to find sum of a and b not sum of mod a and mod b..
Math Expert V
Joined: 02 Sep 2009
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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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Joined: 08 Aug 2018
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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.
Math Expert V
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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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In a Data Sufficiency question if both statements (1) and (2) say the same thing (as in the problem above) doesn't the answer always have to be D or E? You can simplify the problem down to if statement (1) is sufficient then the answer is D and if statement (1) isn't sufficient then the answer is E.

(a) Statement (1) alone is sufficient = If statement (1) is sufficient then statement (2) has to be sufficient since statement (1) and (2) are the same.
(b) Statement (2) alone is sufficient = If statement (2) is sufficient then statement (1) has to be sufficient since statement (1) and (2) are the same.
(c) Both statements together are sufficient, but neither statement alone is sufficient = If statement (1) and (2) are the same then there is no added value to both statements together being sufficient because if one statement alone was sufficient then the other would automatically be sufficient
(d) Each statement alone is sufficient = If statement (1) is sufficient alone then statement (2) is sufficient alone and the answer is satisfied.
(e) statements (1) and (2) together are not sufficient = If statement (1) isn't sufficient then statement (2) isn't sufficient which means that both statements (1) and (2) together are definitely not sufficient so the answer is satisfied.

Am I thinking correctly in saying this?

Thanks!
Math Expert V
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Posts: 59126

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[

Yes, if the statements are saying the same thin the answer must be either D or E.
_________________ Re: M28-04   [#permalink] 10 Nov 2019, 07:39
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# M28-04

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