Probably just the starter solenoid. Easy fix/check on the 18hp Briggs. Follow the large positive wire from the battery, it'll go to the solenoid. Another large wire off that solenoid goes to the starter. Use a large screwdriver to connect those two posts on the solenoid. Connect them quickly or sparks will fly. That will give a direct connection from the battery to the starter. If that works, just replace the solenoid, cheap fix available at the autoparts store (take in the old one to match it up). Ignition key does not have to be on to do this test, or turn it on and the engine will start.
If that doesn't do it and the screwdriver doesn't jump the power, it's the starter itself or a bad ground.
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