[tex]\displaystyle\it\\a+b=2021\bigg(\frac{1}{a}+\frac{1}{b}\bigg)\bigg|:\bigg(\frac{1}{a}+\frac{1}{b}\bigg) \implies (a+b)\cdot\frac{ab}{a+b}=2021 \Leftrightarrow\\ab=2021,~cum~a~si~b~sunt~numere~prime,~am~redus~problema~la~doar\\o~banala~descompunere~in~factori~primi~a~numarului~2021,~avem~ca\\2021=43\cdot47=47\cdot43\implies (a,b)\in\left\{(47,43),(43,47)\right\}.[/tex]