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length of an integer

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Manager
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Kudos [?]: 162 [0], given: 2

length of an integer [#permalink]

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New post 09 Jul 2009, 11:56
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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?

5
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15
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18

[Reveal] Spoiler:
MGMAT/CAT1/Number Properties/Divisibility and Primes/Lengthy Problems

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Kudos [?]: 162 [0], given: 2

Manager
Manager
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Joined: 05 Jun 2009
Posts: 77

Kudos [?]: 3 [0], given: 1

Re: length of an integer [#permalink]

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New post 09 Jul 2009, 12:26
Is there supposed to be an answer posted?

Anyway is there a quicker way to figure this out:

My logic is 2 is the best number to maximize thus so

2*1 = 2
2*2 = 4
...
2*7=128
2*8=256
2*9=512

Options if you maximize X the most you can get is 9 factorials or 512
If you maximize Y the most you can get is 8 factorials or 256
You can't do 9 and 9 so answer 18 is out
And if you do 512 as X and 128 as Y 9 + 7 you can get 16.

Is that even right?

SF

Kudos [?]: 3 [0], given: 1

Senior Manager
Senior Manager
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Joined: 04 Jun 2008
Posts: 286

Kudos [?]: 160 [0], given: 15

Re: length of an integer [#permalink]

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New post 11 Jul 2009, 15:49
agree with sfeiner.

ans should be 16.

take x and y both as 2 to maximize the powers

x^9 = 512 (max possible value since x^10 > 1000)

remaining 999- 512 = 487
3y <= 487
y <= 162
we have to maximize the powers of 2 in y, so y = 2^7 = 128 (y cannot be 2^8 = 256)

so 9 + 7 = 16

Kudos [?]: 160 [0], given: 15

Intern
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Joined: 12 Jul 2009
Posts: 7

Kudos [?]: 1 [0], given: 0

Location: Delhi
Re: length of an integer [#permalink]

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New post 12 Jul 2009, 09:59
Ans .. 16..

2^ 9 + 3 . 2 ^7 < 1000

So..x + y is 9 + 7 = 16.. :lol: :lol:

Kudos [?]: 1 [0], given: 0

Re: length of an integer   [#permalink] 12 Jul 2009, 09:59
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