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number theory senior
Problem
The product of the proper positive integer factors of can be written as , where is the number of positive divisors has, is a positive integer, and the greatest common factor of the three integers , , and is . What is ?
Solution
Recall that by pairing up divisors of , we can show that the product of the positive integer factors of is . We divide this formula by to get the product of the proper positive integer factors of , and we obtain . Therefore, , , and , so .
Final answer
1