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I purchased an e-bike which was advertised as just needing the batteries replaced. The li-ion batteries had been sitting dead for months. Once I got the battery removed it was clear that was not the case. You can see the hole where the plastic melted from this component overheating on the board. The burnt one is the same as those in the center of this photo.
What is it and how do I determine the correct replacement?
You can replace it with any resistor with the same resistance and power rating. It may be easier to solder a larger through hole resistor onto the pads. I think other posters have figured it out, but you can search for resistor code information to decode it yourself. You can buy resistors lots of places, RadioShack was the go to place but I think hardware stores sometimes have them. If you're not in a rush I recommend ordering a kit with a few different kinds.