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

We want to find the number of ordered pairs of natural numbers that have the same, given LCM.

Let N be the value of the given LCM and consider its prime factorization:

or, if the notation for the product is too difficult, use:

where all x are distinct primes and all a are positive integers.

We are looking for the number of possibilities to select the numbers P and Q such that:

Now assume that, in the prime factorization of P, the prime factor xi occurs with exponent j, where:

then in the prime factorization of Q the exponent of xi must be ai.

On the other hand, if exponent of xi in the prime factorization of Q is ai, then its exponent in the factorization of P can be any k such that:

This means that the number of pairs of exponents is:

Since the choices corresponding to different prime factors are independent, the total number of pairs P and Q is:

Example

The least common multiple of two numbers is 675. How many such pairs of numbers are there?

Factor the least common multiple:

Let P and Q be such that:

If the factor 3 appears with an exponent of 1 or 2 in the prime factorization of P, then it must have an exponent of 1 in the factorization of Q. The following table shows only the pairs of powers of 3 that are possible (a total of 7):

For the powers of 3 there are 5 possible pairs:

Therefore the number of choices for P and Q is: