Jack paged through his slide with Jane's remote control.
or
Jack paged through his slide with a Jane's remote control.
Jack paged through his slide s with Jane's remote control. OK or Jack paged through his slide s with a Jane's remote control.
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Jack paged through his slides with Jane's remote control. OK
or
Jack paged through his slides with a Jane's remote control.
.
If Jane has only one remote control:
Jack paged through his slides with Jane's remote control.
If Jane has more than one remote control:
Jack paged through his slides with one of Jane's remote controls.
There is no "a" in either case.
CJ