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This problem trains for: AMC-8, AMC-10, AMC-12.
A number has 120 divisors. The product of all these divisors is:
If a number N has the prime factorization:
Then the number of its positive divisors is:
Let us write explicitly the set of all the divisors as:
The product of all the divisors is:
If the number N is divisible by d, then it is also divisible by:
Therefore, the product of all the divisors can also be written as:
By multiplying the two expressions for the product:
We obtain a general formula for the product of all the divisors of a number:
We can substitute the given number of divisors in this formula to find that: