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PS:Length of a Number

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Manager
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PS:Length of a Number [#permalink]

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New post 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
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Re: PS:Length of a Number [#permalink]

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New post 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
6
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

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Senior Manager
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Re: PS:Length of a Number [#permalink]

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New post 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.

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Re: PS:Length of a Number   [#permalink] 01 Dec 2008, 01:21
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