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# Length of integer

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Manager
Joined: 04 Dec 2008
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Length of integer [#permalink]

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09 Jun 2009, 07:36
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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

Kudos [?]: 225 [0], given: 2

Current Student
Joined: 03 Aug 2006
Posts: 115

Kudos [?]: 304 [0], given: 3

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Schools: Haas School of Business
Re: Length of integer [#permalink]

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09 Jun 2009, 09:33
The answer is C.

We need to find the length of a two digit number, (i.e. a number between 10 and 99) which is the product of 6 prime factors.

Lets start with the smallest prime number i.e. 2 and take the product by multiplying it 6 times by itself

2 x 2 x 2 x 2 x 2 x 2 = 64 < 99 satisfies the criteria.

Now we can increase one of the prime factors to the next larger prime factor

2 x 2 x 2 x 2 x 2 x 3 = 96 < 99 satisfies the criteria

Now if we increase another prime factor to the next larger prime factor

2 x 2 x 2 x 2 x 3 x 3 = 144 > 99 does not satisfy the criteria

Hence the answer is 2 ( the numbers 64 and 96)

Last edited by nookway on 09 Jun 2009, 12:34, edited 1 time in total.

Kudos [?]: 304 [0], given: 3

Intern
Joined: 04 Mar 2008
Posts: 42

Kudos [?]: 3 [0], given: 6

Re: Length of integer [#permalink]

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09 Jun 2009, 09:43
the answer is C as explained above.

Kudos [?]: 3 [0], given: 6

Manager
Joined: 04 Dec 2008
Posts: 104

Kudos [?]: 225 [0], given: 2

Re: Length of integer [#permalink]

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13 Jun 2009, 01:21
nookway wrote:
The answer is C.

We need to find the length of a two digit number, (i.e. a number between 10 and 99) which is the product of 6 prime factors.

Lets start with the smallest prime number i.e. 2 and take the product by multiplying it 6 times by itself

2 x 2 x 2 x 2 x 2 x 2 = 64 < 99 satisfies the criteria.

Now we can increase one of the prime factors to the next larger prime factor

2 x 2 x 2 x 2 x 2 x 3 = 96 < 99 satisfies the criteria

Now if we increase another prime factor to the next larger prime factor

2 x 2 x 2 x 2 x 3 x 3 = 144 > 99 does not satisfy the criteria

Hence the answer is 2 ( the numbers 64 and 96)

Good explanation. I first saw the ques, I didn't know where to start. Thks!

Kudos [?]: 225 [0], given: 2

Re: Length of integer   [#permalink] 13 Jun 2009, 01:21
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# Length of integer

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