version of GTK - by default
Ubuntu 13.10 defaults to Gtk+ 3.8.
dpkg -s libgtk-3-0|grep '^Version' | cut -d' ' -f2-
In Saucy the answer is
The various development packages (the -dev packages) in the repository will also be v3.8.
suggested ways to build 3.10 compatible software
To build GTK3.10 specific packages you'll need to use either the current development version (at the time of writing this) of Ubuntu called 14.04 or use the Gnome-3 PPA
The version reported in 14.04 is 3.10-7-0ubuntu1
My recommendation is to install 14.04 in a virtual machine using a good virtualization software such as Virtual Box or ESXi1.
If you choose to use the Gnome-3 PPA then you need to be aware that not all of Gnome-3 v3.10 is available - so your package your are building may not work.
To add the PPA:
sudo add-apt-repository ppa:gnome3-team/gnome3
sudo apt-get update && sudo apt-get upgrade
Since the PPA updates numerous core packages, this may make your desktop unstable.
Again, my recommendation is to use virtualization software to install a copy on 13.10 upgraded with the Gnome 3 PPA.