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Question-189126




Question Number 189126 by Mingma last updated on 12/Mar/23
Answered by mr W last updated on 12/Mar/23
if 6k≤n<6k+2:  3k+2k+k=n ⇒n=6k ✓  if 6k+2≤n<6k+3:  3k+1+2k+k=n ⇒n=6k+1 →bad  if 6k+3≤n<6k+4:  3k+1+2k+1+k=n ⇒n=6k+2 →bad  if 6k+4≤n<6k+6:  3k+2+2k+1+k=n ⇒n=6k+3 →bad  that means the general solution is  n=6k with k∈Z  for 0<n<2023 there are 337 integers:  n=6,12,18,...,2016,2022
$${if}\:\mathrm{6}{k}\leqslant{n}<\mathrm{6}{k}+\mathrm{2}: \\ $$$$\mathrm{3}{k}+\mathrm{2}{k}+{k}={n}\:\Rightarrow{n}=\mathrm{6}{k}\:\checkmark \\ $$$${if}\:\mathrm{6}{k}+\mathrm{2}\leqslant{n}<\mathrm{6}{k}+\mathrm{3}: \\ $$$$\mathrm{3}{k}+\mathrm{1}+\mathrm{2}{k}+{k}={n}\:\Rightarrow{n}=\mathrm{6}{k}+\mathrm{1}\:\rightarrow{bad} \\ $$$${if}\:\mathrm{6}{k}+\mathrm{3}\leqslant{n}<\mathrm{6}{k}+\mathrm{4}: \\ $$$$\mathrm{3}{k}+\mathrm{1}+\mathrm{2}{k}+\mathrm{1}+{k}={n}\:\Rightarrow{n}=\mathrm{6}{k}+\mathrm{2}\:\rightarrow{bad} \\ $$$${if}\:\mathrm{6}{k}+\mathrm{4}\leqslant{n}<\mathrm{6}{k}+\mathrm{6}: \\ $$$$\mathrm{3}{k}+\mathrm{2}+\mathrm{2}{k}+\mathrm{1}+{k}={n}\:\Rightarrow{n}=\mathrm{6}{k}+\mathrm{3}\:\rightarrow{bad} \\ $$$${that}\:{means}\:{the}\:{general}\:{solution}\:{is} \\ $$$${n}=\mathrm{6}{k}\:{with}\:{k}\in{Z} \\ $$$${for}\:\mathrm{0}<{n}<\mathrm{2023}\:{there}\:{are}\:\mathrm{337}\:{integers}: \\ $$$${n}=\mathrm{6},\mathrm{12},\mathrm{18},…,\mathrm{2016},\mathrm{2022} \\ $$
Commented by Mingma last updated on 13/Mar/23
Excellent!

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