2021 Fall AMC 10B Problems/Problem 6
Problem
The least positive integer with exactly distinct positive divisors can be written in the form
, where
and
are integers and
is not a divisor of
. What is
Solution
Let this positive integer be written as . The number of factors of this number is therefore
, and this must equal 2021. The prime factorization of 2021 is
, so
and
. To minimize this integer, we set
and
. Then this integer is 3^{42} \cdot 2^{46} = 2^4 \cdot 2^{42} \cdot 3^{42} = 16 \cdot 6^{42}
m=16
k=42
m+k = 16 + 42 = 58 = \boxed{B}