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Can anybody tell what the model number of this mower is? I can't find it anywhere. I am looking for a manual so I can work on it.

Did you look under the seat, which is where MTD puts them.

Should be an ID tag under the seat.
The best you will do for a repair manual is the "Must have manual of outdoor Power Equipment " either book 2 or 3.
All over the place as a free download.