It is indeed true that you must be running OS X 10.6.8 or later to install the latest release of iTunes. In turn, you must have the newest iTunes in order to sync your iPhone 5.
Going from OS X 10.5.x to OS X 10.6.x is a paid upgrade. It is no longer officially sold in stores but most customers have been able to get it via telephone.
Call 1-800-MY-APPLE (1-800-692-7753) and tell them you want the Snow Leopard upgrade which they can send on CD for around $20 USD shipped.
Once you have the disc, the upgrade process is fairly simple. After the install, keep running System Preferences > Software Update until no more updates are offered and you should get iTunes 10.7 in the process.
From that point, syncing your iPhones should go smoothly.
As a side note: your Mac is upgradable to the latest version of OS X, 10.8 "Mountain Lion", which is available as a download BUT you must have OS X 10.6.8 installed first. You may wish to do this second upgrade so that your iMac can take advantage of iCloud
syncing with your iPhone 5.
Hopefully, this info will answer your question. If you continue to have questions on this topic, do not rate yet, just reply back to this discussion
. I will be more than happy to work with you further.
Thanks and good luck,
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