2021 AMC 12A Problems/Problem 25
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 is not divisible by 3. Multiply x by 9. Multiplying x by 9 adds a set divisors that are the original divisors multiply by 3 and an additional
divisors that are the originals multiplied by 9 which end up saying that
. Another consequence is multiplying the denominator by
. So f now
so
. A property of multiples of 9 is their digits add up to multiples of 9, so the only possibility is
~Lopkiloinm
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
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