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July 16th, 2013, 11:21 PM  #1 
Newbie Joined: Jul 2013 Posts: 3 Thanks: 0  Number as product of 3 numbers
Hi All, This question might be a bit trivial compared to others being posted here. Still any help would be appreciated. I have a problem staring at me. I have a number say 72. In how many different ways can I express 72 as a product of 3 numbers. I have found a solution to A x B x C =72 which is like (2^a x 3^b), (2^c x 3^d), (2^e x 3^f) should mutiply to 2^3 * 3^2 which implies a+c+e = 3 and b+d+f =2 giving 10 and 6 and totally therefore 60 solutions. But i want to count say 1,1,6 and 6,1,1 and 1,6,1 as just one instead of 3. Can anyone hint at a way? Thanks in advance. Warm Regards, Sankarshan Sridharan 
July 17th, 2013, 02:46 AM  #2 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Number as product of 3 numbers
Consider only products where the factors are in ascending order.

July 17th, 2013, 06:10 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Number as product of 3 numbers
The largest factor must be a divisor of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Go through each of the possibilities: 1: impossible, since then the others must be 1 as well. 2: impossible, since then the others are at most 2 and so the product is at most 8. 3: impossible, since no factor can be composite and so at most you can choose 3 prime factors out of the 5. ... 18: there are two possibilities: 2, 2, 18 and 1, 4, 18. 24: the only possibility is 1, 3, 24. 36: the only possibility is 1, 2, 36. 72: the only possibility is 1, 1, 72. 
July 17th, 2013, 08:02 AM  #4 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Number as product of 3 numbers
What about 1*8*9 and 1*6*12?

July 17th, 2013, 08:05 AM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Number as product of 3 numbers Quote:
 
July 17th, 2013, 08:10 AM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Number as product of 3 numbers
You could check your work with this script (GP, see my .sig): Code: sumdiv(72,a,sumdiv(72/a,b,a>=b&&a*b^2>=72)) Code: sum(a=1,72,sum(b=1,a,sum(c=1,b,a*b*c==72))) 
July 26th, 2013, 02:43 AM  #7 
Newbie Joined: Jul 2013 Posts: 3 Thanks: 0  Re: Number as product of 3 numbers
You can express 72 in different ways like: 72 = 1* 2 * 36 72 = 2 * 3 * 12 72= 3 * 4 * 6 72 = 2 * 4 * 9 72 = 1* 8 * 9 72= 1* 4 * 18 72 = 1 * 6 * 12 and so on. This is a part of factorial equation. The equation can also be done by using Factorial Calculator. 
July 26th, 2013, 02:46 AM  #8 
Newbie Joined: Jul 2013 Posts: 3 Thanks: 0  Re: Number as product of 3 numbers
[quote="james_smith"]You can express 72 in different ways like: 72 = 1* 2 * 36 72 = 2 * 3 * 12 72= 3 * 4 * 6 72 = 2 * 4 * 9 72 = 1* 8 * 9 72= 1* 4 * 18 72 = 1 * 6 * 12 and so on. This is a part of factorial equation. The equation can also be done by visiting the link: http://calculator.mathcaptain.com/facto ... lator.html 
July 27th, 2013, 03:04 PM  #9 
Member Joined: Jul 2013 Posts: 31 Thanks: 0  Re: Number as product of 3 numbers
I think.is the simplest case of a toughest combonotrics problem


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