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For any positive integer n, n>1, the "length" of n is the

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Intern
Joined: 29 Sep 2011
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For any positive integer n, n>1, the "length" of n is the  [#permalink]

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21 Jan 2012, 07:25
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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.

A. 10
B. 9
C. 8
D. 7
E. 6

Thanks!
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Posts: 60647
Re: Length of an integer  [#permalink]

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21 Jan 2012, 07:32
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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.

Check similar questions to practice:
for-any-integer-k-1-the-term-length-of-an-integer-108124.html
for-any-positive-integer-n-the-length-of-n-is-defined-as-126740.html

Hope it helps.
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Manager
Joined: 16 Feb 2012
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Concentration: Finance, Economics
Re: For any positive integer n, n>1, the "length" of n is the  [#permalink]

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10 Feb 2013, 08:39
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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.
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Location: United States
Re: For any positive integer n, n>1, the "length" of n is the  [#permalink]

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21 Oct 2013, 02:37
its really helpful to remember here that 2^10 = 1024 , so the second smallest integer less than 1024 would be 2^9
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For any positive integer n, n>1, the "length" of n is the  [#permalink]

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20 Nov 2014, 01:10
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To have "Maximum length", base should be least.....

$$2^{10} = 1024$$

less than 1000 is $$2^9$$

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Re: For any positive integer n, n>1, the "length" of n is the  [#permalink]

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16 Jul 2017, 17:28
Splendidgirl666 wrote:
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.

A. 10
B. 9
C. 8
D. 7
E. 6

In order to maximize the “length,” we need to minimize the values of the prime factors of n. Since 2 is the smallest prime, let’s see how many factors of 2 we can use to get a product less than 1000. Since 2^9 = 512, we see that the largest possible length of a positive integer less than 1000 is 9.

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Re: For any positive integer n, n>1, the "length" of n is the  [#permalink]

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29 Aug 2019, 09:06
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Re: For any positive integer n, n>1, the "length" of n is the   [#permalink] 29 Aug 2019, 09:06
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