2021 AMC 12A Problems/Problem 25
Contents
Problem
Let denote the number of positive integers that divide
, including
and
. For example,
and
. (This function is known as the divisor function.) Let
There is a unique positive integer
such that
for all positive integers
. What is the sum of the digits of
Solution
Start off with the number x that does not have a factor of 3. Multiply x by 9. Multiplying x by 9 triples the number of divisors and divison by . The number is now
. Multiplying a nonmultiple of 3 by 9 making a bigger f leads to this truth being known,
. A property of multiples of 9 is their digits add up to 9, so the only possibility is
~Lopkiloinm
Note
See problem 1.
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
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