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

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For any positive integer n, the length of n is defined as th [#permalink]

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11 Feb 2010, 16:40
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Question Stats:

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For any positive integer n, the length of n is defined as the 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. None
B. One
C. Two
D. Three
E. Four
[Reveal] Spoiler: OA

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Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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11 Feb 2010, 16:55
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For any positive integer n, the length of n is defined as the 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. None
B. One
C. Two
D. Three
E. Four

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 integer (two digit) 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 $$3$$. $$n=2^p*3^q$$, $$p+q=6$$ max value of $$p$$ and $$q$$ is $$6$$.

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.

With this approach we see that actually $$5<=p<=6$$.

Hope it helps.
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Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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11 Feb 2010, 17:23
That's brilliant!!! I especially love the part where I could take 5 away. This really save tons of time! Thanks!! BTW, Thanks so much for the prompt response!

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Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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11 Feb 2010, 17:25
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smallest value with the given conditions is 2^6 = 64, next one will be 2^5* 3^1 = 96 .. next one will be 100 or greater

Two

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Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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12 Feb 2010, 02:06
Bunuel wrote:
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 integer (two digit) 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 $$3$$. $$n=2^p*3^q$$, $$p+q=6$$ max value of $$p$$ and $$q$$ is $$6$$.

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.

With this approach we see that actually $$5<=p<=6$$.

Hope it helps.

Great explanation.

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22 Apr 2010, 15:53
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The best (quickest) way I can think of to get the answer, is start with 2^6, then move on from there.

2^6=64
2^5*3=96

Obviously 2^5*5 will be more than 2 digits, as will 2^4*3^2. So 64 and 96 are it. Answer is 2 (C).

You may have been looking for something even faster, but this is fast enough for me. Unless someone has a better way.

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24 Aug 2010, 12:30

2^6 = 64
2^5 3 = 96

I think the answer is 2

Posted from my mobile device
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Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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24 Aug 2010, 12:52
Oh great! Now I know how to solve this... Thanks

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Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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15 Oct 2010, 09:04
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only two..
2^6 and 2^5 *3

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Re: 700 Algrbra! Need help again. Thanks so much! [#permalink]

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17 Oct 2010, 19:41
I was not very confident with my approach but got the right answer. i did not rule out 5 the way the our master Bunnel did .

I started with
2^6
2^5*3^1
And then 2^4 onwards nothing fitted.
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30 Nov 2010, 16:07
Guys please I need an explanation!

For any positive n, the length of n is defined as the number of prime factors whose product is n. for example, the length of 75 is 3, since 75= 3x5x5. How many two-digit positive integers have length 6?

-none,-one, -two, -three, -four

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Re: For any positive integer n, the length of n is defined as th [#permalink]

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30 Oct 2014, 10:13
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Hi Bunuel,
Just an FYI the "700+ GMAT Problem Solving Questions with Explanations" word doc has this answer incorrect as B, not c.

Thanks!
Michelle

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For any positive integer n, the length of n is defined as th [#permalink]

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29 Aug 2017, 18:21
YTT wrote:
For any positive integer n, the length of n is defined as the 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. None
B. One
C. Two
D. Three
E. Four

Lets start with smallest prime number $$2$$.

$$2^6 = 64$$ ---------- (Length $$= 6$$)

$$2^7$$ is three digit number hence cannot be $$n$$.

Therefore lets move to next prime number $$3$$.

$$2^5*3^1 = 32*3 = 96$$ ---------- (Length $$= 6$$)

$$2^4*3^2$$ will be three digit number, hence cannot be $$n$$.

Therefore we have $$"Two"$$ two-digit positive integers which have length $$6$$ $$= 64$$ and $$96$$

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Re: For any positive integer n, the length of n is defined as th [#permalink]

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05 Sep 2017, 18:09
YTT wrote:
For any positive integer n, the length of n is defined as the 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. None
B. One
C. Two
D. Three
E. Four

We need to determine how many 2-digit integers have a length of 6, or in other words, how many 2-digit integers are made up of 6 prime factors, not necessarily distinct. Let’s start with the smallest possible numbers:

2^6 = 64 (has a length of 6)

2^5 x 3^1 = 96 (has a length of 6)

Since 2^4 x 3^2 = 144 and 2^5 x 5^1 = 160 are greater than 99, there are no more 2-digit numbers that have a length of 6.

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Re: For any positive integer n, the length of n is defined as th   [#permalink] 05 Sep 2017, 18:09
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