number theory - Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors

Prove that among any 12 consecutive positive integers

there is at least one which is smaller than the sum of
its proper divisors. (The proper divisors of a positive
integer n are all positive integers other than 1 and n
which divide n. For example, the proper divisors of 14 are 2
and 7)

Answer

Hint: Among any $12$ consecutive positive integers, there is one that is a multiple of $12$.

Can you show that $12n$ is smaller than the sum of its divisors for any positive integer $n$?