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What is the greatest value of x such that 2^x is a factor of 768? (A) 16 Jul 2020, 07:44
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5% (low)

Question Stats:

86% (01:21) correct 14% (01:29) wrong based on 155 sessions

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What is the greatest value of x such that \(2^x\) is a factor of 768?

(A) 4

(B) 8

(C) 12

(D) 16

(E) 32
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What is the greatest value of x such that 2^x is a factor of 768? (A) 16 Jul 2020, 08:04
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768 = 3 * 2^8

So, the highest value of x will be 8

Hence, option B

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What is the greatest value of x such that 2^x is a factor of 768? (A) 16 Jul 2020, 09:42
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sjuniv32
What is the greatest value of x such that \(2^x\) is a factor of 768?
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If x is an integer, the answer is 8, since 768 = (3)(256) = (3)(2^8).

But the question doesn't say that x is an integer. The actual answer to the problem is the value of x that makes 2^x exactly equal to 768 (what is known as the base2 logarithm of 768, though you don't need to know anything about logarithms for the GMAT). So the real answer to the question is roughly 9.585. There's a problem with the question - what is the source?
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What is the greatest value of x such that 2^x is a factor of 768? (A) 07 Aug 2020, 16:03
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IanStewart
sjuniv32
What is the greatest value of x such that \(2^x\) is a factor of 768?
But the question doesn't say that x is an integer. The actual answer to the problem is the value of x that makes 2^x exactly equal to 768 (what is known as the base2 logarithm of 768, though you don't need to know anything about logarithms for the GMAT). So the real answer to the question is roughly 9.585. There's a problem with the question - what is the source?
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This is wrong. The question asks, to what power is 2 raised in the prime factorization of 768? An underlying assumption of factorization is that only integers can be called "factors" or "divisible" -- divisible actually means evenly divisible, as stated in any college-level algebra textbook. In fact, prime factorization of a positive integer x means the unique list of prime numbers (integers by definition) raised to positive integer powers whose product represents x.

So since your solution requires 2^(0.585), you haven't done proper prime factorization. 2^(9.585) is definitely not prime factorization of 768, so we say that 768 is not divisible by 2 9.585 times.
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What is the greatest value of x such that 2^x is a factor of 768? (A) 07 Aug 2020, 19:45
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bluejeanbaby

This is wrong. The question asks, to what power is 2 raised in the prime factorization of 768?
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I agree that's probably what the question writer meant to say, but that is not what the question actually says - it says "What is the greatest value of x such that 2^x is a factor of 768?" The greatest factor of 768 is 768. Can 2^x be equal to 768? Yes, when x is approximately 9.585, so that is the answer to the question if x does not need to be an integer, and there's nothing about the wording of the question that requires x to be an integer.

bluejeanbaby

An underlying assumption of factorization is that only integers can be called "factors" or "divisible" -- divisible actually means evenly divisible, as stated in any college-level algebra textbook. In fact, prime factorization of a positive integer x means the unique list of prime numbers (integers by definition) raised to positive integer powers whose product represents x.
Show more

This is all true, at least when you are discussing divisibility over the ring of integers. But 2^x can be a factor of 768 when x is not an integer; all that needs to be true is that 2^x itself is an integer (and a divisor of 768). I do have a Masters degree in Number Theory, so I know something about the subject.
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What is the greatest value of x such that 2^x is a factor of 768? (A) 16 Jan 2026, 08:55
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