how do i update safari 5.1.10 (6534.59.10)

Click here and download El Capitan from the Mac App Store. If you get told that it's incompatible with your computer or you're going to stop upgrading at Lion, choose About this Mac from the Apple menu, check if the computer has at least a Xeon or Core 2 Duo(not Core Duo) CPU and 2GB of RAM, and if it does, order a download code; the online Apple Store also sells download codes for Mountain Lion 10.8, but it has the same system requirements as El Capitan.

Back up your data and check your applications for compatibility before upgrading. In particular, Mac OS X 10.7 and newer don't support PowerPC programs such as versions of Microsoft Office prior to 2008.

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To update Safari, you need to update OS X.