a friend of john's mother---mother's

[LiuJing]

A Chinese English test has this question:

Which is correct, a friend of John's mother or a friend of John's mother's?

It sounds to me the first is better, although not quite grammatical.
Don't native speakers think the first to be more often said?

Thank you.

[tedtmc]

A Chinese English test has this question:

Which is correct, a friend of John's mother or a friend of John's mother's?

It sounds to me the first is better, although not quite grammatical.
Don't native speakers think the first to be more often said?

Thank you.

My friend's father - The father of my friend (not my friend's)
John's mother's friend - A friend of John's mother

not a teacher