mm007 wrote:
If a and b are integers, and a not= b, is \(|a|b > 0\)?
(1) \(|a^b| > 0\)
(2) \(|a|^b\) is a non-zero integer
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.
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