I'm stuck on the following problem:
Find the smallest number $n$ for which the following statement is true: "The number of isomorphism types of abelian groups with $n$ elements is equal to 10". How many such $n$ are there?
My thoughts: I think that the smallest number is $n=144$. How would I prove this? Also, am I correct in saying that there are infinitely many such numbers $n$?
Any help is appreciated!