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July 16th, 2013, 10:21 PM   #1
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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
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July 17th, 2013, 01:46 AM   #2
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Re: Number as product of 3 numbers

Consider only products where the factors are in ascending order.
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July 17th, 2013, 05:10 AM   #3
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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.
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July 17th, 2013, 07:02 AM   #4
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Re: Number as product of 3 numbers

What about 1*8*9 and 1*6*12?
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July 17th, 2013, 07:05 AM   #5
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Re: Number as product of 3 numbers

Quote:
Originally Posted by johnr
What about 1*8*9 and 1*6*12?
1 isn't the largest factor.
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July 17th, 2013, 07:10 AM   #6
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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))
or even
Code:
sum(a=1,72,sum(b=1,a,sum(c=1,b,a*b*c==72)))
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July 26th, 2013, 01:43 AM   #7
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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.
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July 26th, 2013, 01:46 AM   #8
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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
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July 27th, 2013, 02:04 PM   #9
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Re: Number as product of 3 numbers

I think.is the simplest case of a toughest combonotrics problem
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