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For any positive integer n, n>1, the "length" of n is the [#permalink]
21 Jan 2012, 06:25

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30% (01:13) wrong based on 114 sessions

For any positive integer n, n>1, the "length" of n is the number of positive primes (not necessary distinct) whose product is n. For ex, the length of 50 is 3, since 50=2x5x5. What is the greatest possible length of a positive integer less than 1000.

Re: Length of an integer [#permalink]
21 Jan 2012, 06:32

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Expert's post

Splendidgirl666 wrote:

Hi,

is there a short cut for this question:

For any positive integer n, n>1, the "length" of n is the number of positive primes (not necessary distinct) whose product is n. For ex, the length of 50 is 3, since 50=2x5x5. What is the greatest possible length of a positive integer less than 1000.

1. 10 2. 9 3. 8 4. 7 5. 6

Thanks!

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Now, to maximize the length of an integer less then 1,000 we should minimize its prime base(s). Minimum prime base is 2: so 2^x<1,000 --> x<10 --> maximum length is 9 for 2^9=512. Note that 2^9 is not the only integer whose length is 9, for example 2^8*3=768<100 also has the length of 8+1=9.

Re: For any positive integer n, n>1, the "length" of n is the [#permalink]
10 Feb 2013, 07:39

To maximize the length you should use the smallest prime number, 2. 2x2x2x2x2x2x2x2x2 = 2^9 = 512; 2^10 = 1024 which is > 1000, so you have to use 2^9. The answer is B.