Lesson In how many ways the number 27720 can be split into a product of two co-prime factors?

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<H2>In how many ways the number 27720 can be split into a product of two co-prime factors?</H2>

It is proved in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/misc/How-many-subsets-are-there-in-a-given-finite-set-of-n-elements.lesson>How many subsets are there in a given finite set of n elements?</A>&nbsp; under the current topic that the number of all subsets of any finite set

consisting of &nbsp;n&nbsp; elements is &nbsp;{{{2^n}}}.

In this lesson you will find one nice and unexpected application of this theory.

<H3>Problem 1</H3>In how many ways the number &nbsp;27720&nbsp; can be split into the product of two factors what are co-primes?

<B>Solution</B>

27720 = 8*9*5*7*11

is the presentation of the number &nbsp;27720 &nbsp;as the product of five co-prime numbers &nbsp;8, &nbsp;9, &nbsp;5, &nbsp;7, &nbsp;and 11.

So, &nbsp;we have basically &nbsp;5&nbsp; co-prime numbers, &nbsp;and the question is: &nbsp;in how many ways we can collect some of them into the first factor? &nbsp;(Then the rest of them will

automatically go into the second factor). &nbsp;The order of co-primes in the first factor does not make the difference. &nbsp;(Same as the order in the second factor does not).

It is the same as to ask: &nbsp;how many sub-sets are there in the set of &nbsp;5&nbsp; object?

The answer is: &nbsp;{{{2^5}}}.

Indeed, &nbsp;the empty sub-set corresponds to the value &nbsp;1&nbsp; of the first factor.

The sub-sets consisting of &nbsp;1&nbsp; elements, &nbsp;give the values of &nbsp;8, &nbsp;9, &nbsp;5, &nbsp;7, and &nbsp;11 &nbsp;for the first factor.

The sub-sets consisting of 2 elements give the factors &nbsp;8*9, 8*5, 8*7, . . . , 7*11.

The sub-sets . . . . and so on.

Thus the number of ways in which we can construct the first factor is &nbsp;{{{2^5}}}.

But since we do not make the difference between the first and the second factors, &nbsp;we need to divide this number by &nbsp;2.

So, &nbsp;the answer is: &nbsp;The number of ways in which the number &nbsp;27720 &nbsp;can be split into two factors which are co-primes is &nbsp;{{{2^4}}}.

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