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# For any positive integer n, the length of n is defined as

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For any positive integer n, the length of n is defined as [#permalink]  29 Jan 2012, 16:15
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For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.
[Reveal] Spoiler: OA

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Re: 2 digit positive integers with length 6 [#permalink]  29 Jan 2012, 16:21
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enigma123 wrote:
For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.

Basically the length of the integer is the sum of the powers of its prime factors.

Length of six means that the sum of the powers of primes of the two-digit integer must be 6. First we can conclude that 5 can not be a factor of this integer as the smallest integer with the length of six that has 5 as prime factor is 2^5*5=160 (length=5+1=6), not a two-digit integer.

The above means that the primes of the two-digit integers we are looking for can be only 2 and/or 3. $$n=2^p*3^q$$, $$p+q=6$$.

Let's start with the highest value of $$p$$:
$$n=2^6*3^0=64$$ (length=6+0=6);
$$n=2^5*3^1=96$$ (length=5+1=6);

$$n=2^4*3^2=144$$ (length=4+2=6) not good as 144 is a three digit integer.

Questions about the same concept to practice:
length-of-an-integer-126368.html
for-any-integer-k-1-the-term-length-of-an-integer-108124.html

Hope it helps.
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Re: For any positive integer n, the length of n is defined as [#permalink]  29 Jan 2012, 16:46
thanks Bunuel for a very thorough explanation.
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Re: For any positive integer n, the length of n is defined as [#permalink]  30 Jan 2012, 08:01
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Answer is c! 2x2x2x2x2x2 & 2x2x2x2x2x3
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For any positive integer n, the length of n is defined as number [#permalink]  12 Dec 2012, 20:32
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My Solution:

Try increasing prime numbers with length 6:
Trial 1: $$2^6=64$$ Valid
Trial 2: $$3^6 =729$$ Invalid

This means our candidate 2-digit numbers have combinations of $$2$$ and $$3$$

$$2^6=64$$
$$2^5x3^1=96$$
$$2^4x3^2=144$$ Invalid

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For any positive integer n, the length of n is defined as number [#permalink]  10 Feb 2013, 02:35
2x2x2x2x2x2 = 2^6 = 64
2x2x2x2x2x3 = 2^5x3 = 96

These are only two possible solutions, therefore the answer is C.
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Re: For any positive integer n, the length of n is defined as [#permalink]  08 Mar 2013, 15:13
Try the smallest possible value first: In this case it is 2^6 which equals 64.

If we replace the last 2 with 3, then we have 2^5*3 = 96

From here we can positively assume that any other number will have more than 2 digits. So the answer is (C) 2 numbers that have length 6 and are only 2 digits.
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Re: For any positive integer n, the length of n is defined as [#permalink]  09 Mar 2014, 00:18
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Re: For any positive integer n, the length of n is defined as [#permalink]  19 Apr 2014, 23:45
Length of a number = Total number of prime factors of the number.

Any composite number can be represented as a product of prime numbers as shown below:

N= 2^x * 3^y * 5^z * 7^a ...so on

Since our requirement is a two digit number, we shall raise maximum power for smallest prime factor.

N= 2^6 = 64 has Length 6

N= 2^5 * 3 = 96 has length 6

N=2^4 * 3^2 = 144 is a three digit number and so on the other combinations would reveal 3 digit numbers.

Hence there are only 2 numbers
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Re: For any positive integer n, the length of n is defined as [#permalink]  30 Apr 2014, 23:48
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Just writing it out took me .46 sec.:

length of 6, lets take the lowest prime factor 2

2x2x2x2x2x2 = 64

Now substitute the last 2 by a 3, and see the solution gets 96. We can think of what will happen when we substitute another 2 for a three.

Hence, C (2)
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Re: For any positive integer n, the length of n is defined as [#permalink]  30 May 2015, 02:39
Hello from the GMAT Club BumpBot!

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Re: For any positive integer n, the length of n is defined as   [#permalink] 30 May 2015, 02:39
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