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The iPhone 5 requires iTunes 10.7 or later, which require Mac OS X 10.6.8 or later.
First, check that your Mac is compatible > http://support.apple.com/kb/sp575 Then, buy Snow Leopard > http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard#
Make a backup of your data with Time Machine, Carbon Copy Cloner or Disk Utility's Restore feature, insert the DVD and upgrade. After upgrading, open > Software Update, and install OS X 10.6.8. Finally, run Software Update again and install iTunes 11.0.4.
If you want iTunes 10.7 instead of 11, download it from the Apple website > http://support.apple.com/kb/dl1576