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PS:Length of a Number

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Manager
Joined: 08 Aug 2008
Posts: 228

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01 Dec 2008, 00:53
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For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?
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Director
Joined: 14 Aug 2007
Posts: 702
Re: PS:Length of a Number [#permalink]

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01 Dec 2008, 01:12
prasun84 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?
5
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15
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Old yet nice problem.

the trick here is "maximum" length.
Number 2 being the smallest prime number can help us obtain maximum length.
2^9 = 512 (x)

again if we look at highest power of 2,
1)it should be lesser than 999-512 = 487
AND
2)it also has to be 3*somenumber

so we have 2^7 = 128 that satisfies both these conditions.

thus we have 9+7 = 16 as the max length of x+3y<1000
Senior Manager
Joined: 28 Feb 2007
Posts: 296
Re: PS:Length of a Number [#permalink]

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01 Dec 2008, 01:21
D.
max length for x and y can be achieved when they contain max number of the smallest prime # (2).
so x=512=2^9 and y=128=2^7.
512+3*128<1000
9+7=16.

--== Message from GMAT Club Team ==--

This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Re: PS:Length of a Number   [#permalink] 01 Dec 2008, 01:21
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