Is the 'application of binomial theorem to find the remainder ’ part of the GRE quants syllabus? has anyone encountered such questions in practice, or on an actual test where binomial theorem should be used?
No it is out of the scope of GRE.
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Link to all the concepts that GRE test for Quant Part: https://www.ets.org/gre/revised_general/prepare/quantitative_reasoning/
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