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# Number theory.....

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Number theory..... [#permalink]  05 Oct 2010, 04:39
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K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K, how many factors does K have?
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Re: Number theory..... [#permalink]  05 Oct 2010, 05:15
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pzazz12 wrote:
K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K, how many factors does K have?

I assume it's K=w*x*y*z, where w, x, y, z are prime numbers.

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.

So if K=w^1*x^1*y^1*z^1, where w, x, y, z are prime numbers then number of factors of K including 1 and K itself would be (1+1)(1+1)(1+1)(1+1)=16 and excluding 1 and K: 16-2=14.

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

Hope it helps.
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Re: Number theory..... [#permalink]  05 Oct 2010, 05:40
^^^ Awesome Bunuel. I was clueless when saw this question. Thanks. Thank you also for the correction.
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Re-taking GMAT. Hope the charm works this time..

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Re: Number theory..... [#permalink]  05 Oct 2010, 06:23
If you don't know the formula you can get to it the following way:

Suppose w = 2, x = 3, y = 5 and z = 7. Then wxyz = 210.

To find out how many factors 210 has, go through different combinations of the prime factors as follows. You don't even need to calculate the answers but I have done so here:

(factor #1) 2
(factor #2) 3
(factor #3) 5
(factor #4) 7

(factor #5) 2 x 3 = 6
(factor #6) 2 x 5 = 10
(factor #7) 2 x 7 = 14
(factor #8) 2 x 3 x 5= 30
(factor #9) 2 x 3 x 7 = 42
(factor #10) 2 x 5 x 7 = 70

(factor #11) 3 x 5 = 15
(factor #12) 3 x 7 = 21
(factor #13) 3 x 5 x 7 = 105

(factor #14) 5 x 7 = 35

Since we are not including 1 and 210, answer = 14
Re: Number theory.....   [#permalink] 05 Oct 2010, 06:23
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