plural distributive

Can anyone help explain these sentences as far as plural distributive is concerned?

1. All doctors should carry a pager. Why 'a pager' when 'All doctors' as the subject is plural? Why is the object 'pager' not plural too?

2. All children should blow their noses. Why 'noses' and not a nose? Is it because nose is a part of human body and thus, it is should be plural??

Please clarify me on this.

thanks.

1. 

'All doctors' is a generalisation. It means 'Each doctor'. So strictly it's not a plural!

2.

This should read:

All children should blow their nose.

This is in line with the first statement.

I know some people will say 'blow their noses' and I am not going to say 'they are wrong' I'm afraid English grammar is not an exact science!