2021 Fall AMC 12B Problems/Problem 12
Contents
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
so
so the difference is
~lopkiloinm
Solution 2
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
, inclusive. So, the sum of all of the divisors of
is just
. Therefore,
. Similarly, since
,
. Therefore, the answer is
.
~mahaler
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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