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K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K

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K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K [#permalink] New post 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: K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K [#permalink] New post 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: K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K [#permalink] New post 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: K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K [#permalink] New post 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
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Re: K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K [#permalink] New post 05 Jul 2015, 06:44
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Re: K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K [#permalink] New post 05 Jul 2015, 08:24
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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?


You need more information to answer the question. If w, x, y and z are distinct primes, then wxyz will have 16 positive factors in total (including 1 and wxyz), as Bunuel explained above. But the question doesn't say that the primes are distinct, and if two or more of the primes are equal, then wxyz will definitely have fewer than 16 positive factors.
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Re: K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K   [#permalink] 05 Jul 2015, 08:24
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