If a and b are integers, and a not= b, is |a|b > 0? (1)

my answer is E

Given: a , b are ints. and a is different from b
asking: |a| * b > 0


what the question is really asking if b > 0 [ |a| is always >0 ]

(1) |a^b| >0
----------------
says nothing, cuz |x| is always > 0
statement 1 is insufficient

(2) |a|^b is not zero
-------------------------
also says nothing .. we know |a|^b > 0
b could be -ve or +ve
statement 2 is insufficient

(1) and (2) together
------------------------
both statements really say nothing about b

final answer is E

>>Please tell me how can |a| be taken as positive in the above steps without knowing its sign..I mean if a is negative,then |a| wil be negative..Right?Am i missing anythin badly?

Hi Bunuel,
I've always struggled when to consider 0 as an integer and when not. Is there any concept that you can share? Appreciate your help!!

0 is neither positive nor negative even integer.

Check for more here: number-properties-tips-and-hints-174996.html

OFFICIAL SOLUTION

Let us start be examining the conditions necessary for \(|a|b > 0\). Since |a| cannot be negative, both |a| and b must be positive. However, since |a| is positive whether a is negative or positive, the only condition for a is that it must be non-zero.

Hence, the question can be restated in terms of the necessary conditions for it to be answered "yes":

“Do both of the following conditions exist: a is non-zero AND b is positive?”

(1) INSUFFICIENT: In order for a = 0, \(|a^b|\) would have to equal 0 since 0 raised to any power is always 0. Therefore (1) implies that a is non-zero. However, given that a is non-zero, b can be anything for \(|ab| > 0\) so we cannot determine the sign of b.

(2) INSUFFICIENT: If a = 0, |a| = 0, and \(|a|^b = 0\) for any b. Hence, a must be non-zero and the first condition (a is not equal to 0) of the restated question is met. We now need to test whether the second condition is met. (Note: If a had been zero, we would have been able to conclude right away that (2) is sufficient because we would answer "no" to the question is |a|b > 0?) Given that a is non-zero, |a| must be positive integer. At first glance, it seems that b must be positive because a positive integer raised to a negative integer is typically fractional (e.g., \(a^{-2} = \frac{1}{{a^2}}\). Hence, it appears that b cannot be negative. However, there is a special case where this is not so:

If |a| = 1, then b could be anything (positive, negative, or zero) since \(|1|^b\) is always equal to 1, which is a non-zero integer . In addition, there is also the possibility that b = 0. If |b| = 0, then \(|a|^0\) is always 1, which is a non-zero integer.

Hence, based on (2) alone, we cannot determine whether b is positive and we cannot answer the question.

An alternative way to analyze this (or to confirm the above) is to create a chart using simple numbers as follows:

a b Is \(|a|^b\) a non-zero integer? Is \(|a|b > 0\)?
1 2 Yes Yes
1 -2 Yes No
2 1 Yes Yes
2 0 Yes No

We can quickly confirm that (2) alone does not provide enough information to answer the question.

(1) AND (2) INSUFFICIENT: The analysis for (2) shows that (2) is insufficient because, while we can conclude that a is non-zero, we cannot determine whether b is positive. (1) also implies that a is non-zero, but does not provide any information about b other than that it could be anything. Consequently, (1) does not add any information to (2) regarding b to help answer the question and (1) and (2) together are still insufficient. (Note: As a quick check, the above chart can also be used to analyze (1) and (2) together since all of the values in column 1 are also consistent with (1)).

The correct answer is E.

Hi Bunuel,

For statement 2, how can "b" be negative? Wouldn't a negative value make it a fraction? Given it's a non-zero integer, doesn't "b" have to be >=0?

a = 1 and b = -1;
a = 1 and b = -2;
a = 1 and b = -3;
...

a = -1 and b = -2;
a = -1 and b = -3;
a = -1 and b = -4;
...

Bunuel

Please advise me on second statement b can not be negative as if b is negative then |a|^b let say |2|^-3 = 1/8 not equal to an integer. Please advise where i am going wrong about this.

Posted from my mobile device

The solution says, IF a = 1, then b can be any integer, positive or negative. For example, |1|^(-3) = 1. So, in case a = 1 (or a = -1), b can be any positive or negative integer or 0.

If |a| > 1, for example, if a = 2, -2, 3, -3, ... then yes, b must be nonnegative integer: 0, 1, 2, 3, ...

Hope it helps.

>>Please tell me how can |a| be taken as positive in the above steps without knowing its sign..I mean if a is negative,then |a| wil be negative..Right?Am i missing anythin badly?[/quote]

Absolute value of of an expression is alway non-negative: \(|some \ expression|\geq{0}\). Please check Walker's post on Absolute Value at: https://gmatclub.com/forum/math-absolut ... 86462.html

As for the question:

If a and b are integers, and a does not equal to b, is |a|*b > 0?
(1) |a^b| > 0
(2) |a|^b is a non-zero integer.

\(|a|*b>0\) is true when \(b>0\) and \(a\) does not equal to zero.

(1) \(|a^b| > 0\) --> \(a\) does not equal to zero, but we don't know about \(b\), it can be any value, positive or negative. Not sufficient.

(2) \(|a|^b\) is a non-zero integer --> \(a\) can be 1 and \(b\) any integer, positive or negative. Not sufficient.

(1)+(2) If \(a=1\) and \(b=2\), then \(|a|*b>0\), but if \(a=1\) and \(b=-2\), then \(|a|*b<0\). Not sufficient.

Answer: E.

Other discussion of this question at: https://gmatclub.com/forum/good-set-of-ds-85413.html
Similar question: https://gmatclub.com/forum/the-power-of ... 01661.html

Hope it helps.[/quote]

Bunuel

Please advise me on second statement b can not be negative as if b is negative then |a|^b let say |2|^-3 = 1/8 not equal to an integer. Please advise where i am going wrong about this.

Posted from my mobile device[/quote]

The solution says, IF a = 1, then b can be any integer, positive or negative. For example, |1|^(-3) = 1. So, in case a = 1 (or a = -1), b can be any positive or negative integer or 0.

If |a| > 1, for example, if a = 2, -2, 3, -3, ... then yes, b must be nonnegative integer: 0, 1, 2, 3, ...

Hope it helps.[/quote]

Thanks 🙏 a lot

Bunuel can you advise best source to learn more about absolute values.

10. Absolute Value

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.