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This essential trains for: AMC-8, AMC-10, AMC-12, AIME, Math Kangaroo 7-8, Math Kangaroo 9-10.

If we multiply all the divisors of a number, what product do we get?

The first observation is that, by multiplying divisors of a number, no prime numbers can be factors of the product except the prime factors of the number itself.

If the number is composite, it can be factored into primes uniquely as:

equation

where:

equation

are k prime numbers ordered increasingly (so as to make sure they are all different and all the multiple occurences of a given prime in the factorization have been accounted for by using integer exponents)

and:

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We also know, from a previous lesson, that the total number of positive divisors of N is:

equation

Let us consider the set of all the positive factors/divisors of N:

equation

For any factor f of the number N, there is another factor of N, f' such that:

equation

Since F(N) is the set of all the positive divisors, it is true that the primed factors are only a rearrangement of this set:

equation

By multiplying each f with its corresponding f' we get:

equation

Since there are τ factors, by multiplying them all we get:

equation

We get that:

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Since the set of factors is the same, and denoting the product of all the factors with P(N):

equation

and, finally:

equation

Example: By multiplying all the positive divisors of the number:

equation

we obtain the number M. How does M change if N is squared?

Since:

m30

equation

equation

equation

equation

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If N is squared, M is raised to the power:

equation