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

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

It is proved in the lesson How many subsets are there in a given finite set of n elements? under the current topic that the number of all subsets of any finite set
consisting of n elements is 2%5En

.

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

Problem 1

In how many ways the number 27720 can be split into the product of two factors what are co-primes?

Solution

27720 = 8*9*5*7*11

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

So, we have basically 5 co-prime numbers, and the question is: in how many ways we can collect some of them into the first factor? (Then the rest of them will
automatically go into the second factor). The order of co-primes in the first factor does not make the difference. (Same as the order in the second factor does not).

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

The answer is: 2%5E5

.

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

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

The sub-sets consisting of 2 elements give the factors 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 2%5E5

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

So, the answer is: The number of ways in which the number 27720 can be split into two factors which are co-primes is 2%5E4

.

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