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# “length of an integer”

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Intern
Joined: 29 Apr 2009
Posts: 1

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18 Jul 2009, 18:05
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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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Senior Manager
Joined: 18 Jun 2009
Posts: 351
Location: San Francisco
Re: “length of an integer” [#permalink]

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18 Jul 2009, 21:47
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KUDOS
Wow, nice question thought took somewhat like 5 minutes

So it says maximum number of prime numbers, so I thought it should have maximum number of 2's to have maximum length

So I choose x = 2^9 = 512

Subtracting from 1000 we get 488

Y = the least number which is a power of 2 but less than 1/3rd of 488 = 2^7

therefore Y = 2^7 = 128 (note 3Y < 488 )

therefore we have the length as 9 + 7 = 16

Manager
Joined: 27 Jun 2008
Posts: 144
Re: “length of an integer” [#permalink]

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21 Jul 2009, 03:17
yeah you are right, I think I have seen it earlier in OG or QG.
Senior Manager
Joined: 23 Jun 2009
Posts: 355
Location: Turkey
Schools: UPenn, UMich, HKS, UCB, Chicago
Re: “length of an integer” [#permalink]

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21 Jul 2009, 05:18
Very nice solution.
Re: “length of an integer”   [#permalink] 21 Jul 2009, 05:18
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