The entire B737Max family (B737Max7,8,9,10) uses CFM International LEAP-1B (Source) engines, despite the B737Max10 carrying up to 60 more people than Max8.

My question is how do all of the aircraft in the family achieve the same takeoff performance, despite such a remarkable difference in their size, with the same engines?

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    $\begingroup$ Do you have a source that says they will achieve the same takeoff performance? I assume you are talking about takeoff distance required? $\endgroup$ – Ben Jun 19 '17 at 11:15

There are several model variations of that type engine.

Refer to the Type Certificate Data Sheet (TCDS) for the Boeing 737. You will find listed the specific model number numbers and thrust for each approved engine.

For example, here are the CFM engine models approved for specific model/serial numbers of the 737. You will have to get the latest revision of the TCDS for new models, like the 737 MAX. This excerpt from the 737 TCDS shows the model variety of one type engine (the CFM56-7B) as approved for certain 737s:

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