A quick Google search of "cub cadet 1782 owner's manual" produced this site:
http://ccmanuals.info/pdf/1782%20Owners%20Manual.pdf
Page 42 shows the battery spec; amperage cranking power range.
But I would suspect any 12v garden/utility battery at the auto parts store would be able to work. If you cannot find something, don't underestimate the power of a simple Google search. 
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Stanton
1980 IH 782, Kohler M18
IH #1 Cart, original
IH 42" Blade, modified to 50" and hydraulic
2015 XT2, Kawasaki 23 HP
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