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If a and b are integers, is a^5 < 4^b ?

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If a and b are integers, is a^5 < 4^b ?

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If a and b are integers, is a^5 < 4^b ?

(1) a^3 = -27
(2) b^2 = 16

A

Source: Official Guide 2020

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4^b is always positive, no matter what b is. If a^3 = -27, then a is negative and so is a^5. So it's certainly true that a^5 < 4^b, because negative numbers are smaller than positive numbers. So Statement 1 is sufficient.

Statement 2 is not sufficient; while we know b is either 4 or -4, we have no idea what a is. If a is negative, then the answer to the question will be 'yes', but if a = 1,000,000,000, the answer will be 'no'.

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AbeNeedsAnswers wrote:
If a and b are integers, is a^5 < 4^b ?

(1) a^3 = -27
(2) b^2 = 16

A

Source: Official Guide 2020
\

Target question: Is a^5 > 4^b

Statement 1: a³ = -27
Solve to get: a = -3
So, a^5 = (-3)^5 = -243
Since 4^b will be POSITIVE for all values of b, the answer to the target question is NO, a^5 is definitely NOT greater than 4^b
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: b^2 = 16
Solve to get: EITHER b = 4 OR b = -4
Let's test some possible cases:
Case a: a = 1 and b = 4. In this case, a^5 = 1^5 = 1, and 4^b = 4^4 = 256. Here, the answer to the target question is NO, a^5 is definitely NOT greater than 4^b
Case b: a = 10 and b = 4. In this case, a^5 = 10^5 = 100,000, and 4^b = 4^4 = 256. Here, the answer to the target question is YES a^5 is greater than 4^b
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

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Hi All,

We're told that A and B are INTEGERS. We're asked if A^5 is less than 4^B. This question can be approached with a mix of Number Properties and TESTing VALUES. To start, it's worth noting that raising +4 to ANY power will lead to a POSITIVE value (for example, 4^0 = 1, 4^-1 = 1/4, etc.).

(1) A^3 = -27

Fact 1 tells us that A = -3, although once you realize that A is a NEGATIVE value, you can stop working. By extension, A^5 would be a NEGATIVE value - and since 4^B is a POSITIVE value, we know that A^5 will ALWAYS be less than 4^B. Thus the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

(2) B^2 = 16

Fact 2 tells us that B = 4, but we don't know anything about the value of A. 4^4 = 16^2 = 256
IF...
A = 1, then A^5 = 1 and the answer to the question is YES.
A = 4, then 4^5 is clearly GREATER than 4^4, so the answer to the question is NO.
Fact 2 is INSUFFICIENT

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

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