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# M01 #35 clarification

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Manager
Joined: 13 Jul 2010
Posts: 165

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31 Oct 2010, 12:32
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Find the number of divisors of 90 and then the number of divisors of that result.

The following formula can be used to find the number of divisors of any given number. Factor 90 ( $$90= 2*3^2*5$$ ) and then multiply the powers+1 $$2*3*2=12$$ . Therefore, $$@90=12$$ .

Another approach is to write out all of the distinct divisors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90; the answer is 12, not 10 because 1 and 90 are also included.

The explanation says there is a formula to determine the number of divisors rather than counting them out, I can't follow the formula can the club please define if your familiar with it? Thanks

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Joined: 02 Sep 2009
Posts: 41601

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31 Oct 2010, 12:36
gettinit wrote:
Find the number of divisors of 90 and then the number of divisors of that result.

The following formula can be used to find the number of divisors of any given number. Factor 90 ( $$90= 2*3^2*5$$ ) and then multiply the powers+1 $$2*3*2=12$$ . Therefore, $$@90=12$$ .

Another approach is to write out all of the distinct divisors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90; the answer is 12, not 10 because 1 and 90 are also included.

The explanation says there is a formula to determine the number of divisors rather than counting them out, I can't follow the formula can the club please define if your familiar with it? Thanks

Discussed here: m01-78589.html?hilit=finding%20factors#p786451

MUST KNOW FOR GMAT:

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:

If $$@x$$ is the number of distinct positive divisors of $$x$$ , what is the value of $$@@90$$?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

The question defines $$@x$$ as the number of distinct positive divisors of $$x$$. Say $$@6=4$$, as 6 have 4 distinct positive divisors: 1, 2, 3, 6.

Question: $$@@90=?$$

$$90=2*3^2*5$$, which means that the number of factors of 90 is: $$(1+1)(2+1)(1+1)=12$$. So $$@90=12$$ --> $$@@90=@12$$ --> $$12=2^2*3$$, so the number of factors of 12 is: $$(2+1)(1+1)=6$$.

For more on these issues check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
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04 Nov 2010, 18:24
Thanks Bunuel, very helpful and really appreciate the responsiveness!

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Re: M01 #35 clarification   [#permalink] 04 Nov 2010, 18:24
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# M01 #35 clarification

Moderator: Bunuel

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