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

Question

K = wxyz, where w, x, y, z are prime numbers. Not including 1 and K, how many factors does K have?

Bunuel · Accepted Answer

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.

piyushagarwal · Answer

^^^ Awesome Bunuel. I was clueless when saw this question. Thanks. Thank you also for the correction.

Marshmallow · Answer

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

IanStewart · Answer

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.

CounterSniper · Answer

let k =2*3*5*7
total factors excluding 1 and k = (2*2*2*2)-2=14

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards