2021 Fall AMC 12B Problems/Problem 12
Problem
For a positive integer, let be the quotient obtained when the sum of all positive divisors of n is divided by n. For example, What is
Solution 1
The prime factorization of is and the prime factorization of is Note that Therefore, the answer is ~lopkiloinm ~MRENTHUSIASM
Solution 2
Let denotes the sum of the positive integer divisors of so
Suppose that is the prime factorization of Since is multiplicative, we have The prime factorization of is and the prime factorization of is Note that Therefore, the answer is ~MRENTHUSIASM
Solution 3
We see that the prime factorization of is Each of its divisors is in the form of or for a nonnegative integer We can use this fact to our advantage when calculating the sum of all of them. Notice that is the sum of the two forms of divisors for each from to inclusive. So, the sum of all of the divisors of is just Therefore, we have Similarly, since we have
Finally, the answer is
~mahaler
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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All AMC 12 Problems and Solutions |
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