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For any positive integer n, n>1, the "length" of n is the
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21 Jan 2012, 06:25
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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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Re: Length of an integer
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21 Jan 2012, 06:32
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. Answer: B. Check similar questions to practice: foranyintegerk1thetermlengthofaninteger108124.htmlforanypositiveintegernthelengthofnisdefinedas126740.htmlHope it helps.
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Re: For any positive integer n, n>1, the "length" of n is the
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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.
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Re: For any positive integer n, n>1, the "length" of n is the
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21 Oct 2013, 01:37
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
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20 Nov 2014, 00:10
To have "Maximum length", base should be least..... \(2^{10} = 1024\) less than 1000 is \(2^9\) Answer = B = 9
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Re: For any positive integer n, n>1, the "length" of n is the
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16 Jul 2017, 16:28
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. Answer: B
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Re: For any positive integer n, n>1, the "length" of n is the
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13 Aug 2018, 20:47
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