2020 AMC 12B Problems/Problem 24
Let denote the number of ways of writing the positive integer as a productwhere , the are integers strictly greater than , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number can be written as , , and , so . What is ?
Solution 1
To make a product of , we can either have just , or we can have a divisor of , , followed by a way to make a product of . Thus, for all possible . It's easy to calculate for all powers of , since powers of only have powers of as divisors. We have Now we will calculate for all in the form , for . Note that each divisor of is either of the form or . Therefore, to calculate each , we will sum the first values in both the tables we created, and add . We have
Solution 2
Bash. Since , for the number of , we have the following cases:
Case 1: , we have , only 1 case.
Case 2: , we have , totally cases.
Case 3: , we have , totally cases.
Case 4: , we have , totally cases.
Case 5: , we have , totally cases.
Case 6: , we have , totally cases.
Thus, add all of them together, we have cases. Put .
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See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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