How many combinations of 3 numbers can be used to calculate 1 number

[Optic Aeon]

Hi all,

I'm relatively new to experiencing the JOY of math.

Can someone tell me the steps for how I would determine how many variables of 3 numbers could multiply to produce 27000

For instance, I know 50*20*27 = 27000

But I need to know how many other other combinations of 3 numbers can be multiplied to get the result of 27000.

Thanks!

 

[tkhunny]

You'll need a prime factorization of 27,000. Please demonstrate.

 

[Optic Aeon]

You'll need a prime factorization of 27,000. Please demonstrate.

I want to know all possible number combos, not only primes

 

[Dr.Peterson]

Can someone tell me the steps for how I would determine how many variables of 3 numbers could multiply to produce 27000

For instance, I know 50*20*27 = 27000

But I need to know how many other other combinations of 3 numbers can be multiplied to get the result of 27000.

First, you have to clearly define the problem!

I suspect that by "number" you mean "positive integer", and are not allowing them to be fractions, or negative numbers, or worse. That needs to be clarified. Can any of them be 1?

Then, by "combination", shall we assume you don't consider order to matter, so that 20*50*27 wouldn't be different?

By "3 numbers", do you mean they must be different numbers, so you would exclude 30*30*30?

These are the sorts of things a "joyful mathematician" will think of first!

Then start with a smaller number to develop a method. How many sets of 3 numbers can you multiply to make, say, 50? List them out first, then look for a way to count them without listing. That will definitely involve factoring.

And the fun part of all this is, of course, figuring it out for yourself, so none of us (I hope) will give you detailed steps!

 

[tkhunny]

I want to know all possible number combos, not only primes

I would start with a prime factorization. Please demonstrate.

 

[Steven G]

I want to know all possible number combos, not only primes

Yes we understand that but you need to understand that 3 positive integers that multiply out to 2700 will use ONLY the prime factors of 2700 and 1, if allowed

 

[pka]

I'm relatively new to experiencing the JOY of math.
Can someone tell me the steps for how I would determine how many variables of 3 numbers could multiply to produce 27000
For instance, I know 50*20*27 = 27000
But I need to know how many other other combinations of 3 numbers can be multiplied to get the result of 27000.

Several have asked about the factorization of \(27000=2^3\cdot 3^3\cdot 5^3\)
Here is one number combination: \(\underbrace {(2 \cdot {3^2} \cdot 5)}_{90}\underbrace {({2^2} \cdot 3)}_{12}\underbrace {({5^2})}_{25} = 27000\)
The task is to find all possible combinations of three using powers of the three primes.
It is not clear if one of the three can be \(1\) or even say \(1\cdot 2\cdot 13500\)