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 is divided by For example, What is
Solution 1
The prime factorizations of and are and respectively. Note that is the sum of all fractions of the form where is a positive divisor of By geometric series, it follows that Therefore, the answer is
~lopkiloinm ~MRENTHUSIASM
Solution 2
The prime factorization of is so each of its positive divisors is of the form or for some integer such that We will use this fact to calculate the sum of all its positive divisors. Note that is the sum of the two forms of positive divisors for all such By geometric series, the sum of all positive divisors of is from which Similarly, since the prime factorization of is we have
Therefore, the answer is
~mahaler
Solution 3
Let denotes the sum of all positive divisors of so
Suppose that is the prime factorization of Since is multiplicative, we have by geometric series.
The prime factorizations of and are and respectively. Note that Therefore, the answer is
~MRENTHUSIASM
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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