MathDB

Problem 9

Part of 2019 LIMIT Category C

Problems(2)

divisibility statements

Source: LIMIT 2019 CCS1 P9

4/28/2021
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}
number theory
idempotent matrices

Source: LIMIT 2019 CCS2 P9

4/28/2021
P∈An(R)={Mn×n∣M2=M}P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}. Which of the following are true? <spanclass=′latex−bold′>(A)</span> PT=P,∀P∈An(R)<span class='latex-bold'>(A)</span>~P^T=P,\forall P\in A_n(\mathbb R) <spanclass=′latex−bold′>(B)</span> ∃P≠0,P∈An(R) with tr⁡(P)=0<span class='latex-bold'>(B)</span>~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0 <spanclass=′latex−bold′>(C)</span> ∃Xn×r such that Px=X for r=rank⁡(P)<span class='latex-bold'>(C)</span>~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)
Matriceslinear algebramatrix