[tex]\displaystyle\it\\3)~S=7^{2018}+4^{2019}+5^{2020}=7^{\mathcal{M}_{4}+2}+4^{\mathcal{M}_2} + 5^{2020}=\\\overline{...9}+\overline{...6}+\overline{...5}=\boxed{\it\overline{...0}}.\\----------------\\4)~\overline{abc}-\overline{ab}=111 \Leftrightarrow 10\overline{ab}+c-\overline{ab}=9\overline{ab}+c=111,~9\overline{ab}\leq 111 \implies \\a\leq 1,~de~unde~\boxed{\it a=1}.\\revenim,~90+9b+c=111 \implies 9b+c=21 \implies[/tex]
[tex]\displaystyle\it[/tex][tex]\displaystyle\it\\9b=21-c,~21-c=\mathcal{M}_9,~21-c\geq 12\implies 21-c=18 \implies \boxed{\it c=3}\\\implies \boxed{\it b=2},~deci~\boxed{\it \overline{abc}=123}.\\----------------\\5)~descompunem~numarrul~810000~in~factori~primi.\\810000=81\cdot10^4=3^4\cdot2^4\cdot5^4.\\2^x\cdot3^{x+1}\cdot5^{x+2}\cdot2^{y+2}\cdot3^{y+1}\cdot5^y=2^4\cdot3^4\cdot5^4 \implies\\2^{x+y+2}\cdot3^{x+y+2}\cdot5^{x+y+2}=2^4\cdot3^4\cdot5^4 \implies\\x+y+2=4 \implies \boxed{\it x+y=2}.[/tex]
