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divisibility statements

Source: LIMIT 2019 CCS1 P9

April 28, 2021
number theory

Problem Statement

Which of the following are true? <spanclass=′latex−bold′>(A)</span> For every n∈N,n3−n is divisible by 6<span class='latex-bold'>(A)</span>~\text{For every }n\in\mathbb N,n^3-n\text{ is divisible by }6 <spanclass=′latex−bold′>(B)</span> For every n∈N,n7−n is divisible by 42<span class='latex-bold'>(B)</span>~\text{For every }n\in\mathbb N,n^7-n\text{ is divisible by }42 <spanclass=′latex−bold′>(C)</span> Every perfect square is of the form 3m or 3m+1 for some n∈N<span class='latex-bold'>(C)</span>~\text{Every perfect square is of the form }3m\text{ or }3m+1\text{ for some }n\in\mathbb N <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
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