E.698. Determinați numerele naturale abcab‾\overline{abcab}abcab scrise în baza 10,10,10, divizibile cu 913.913.913.
Răspuns: abcab‾=83083.\overline{abcab}=83083.abcab=83083.
abcab‾=1000⋅ab‾+100c+ab‾=1001⏟7⋅11⋅13⋅ab‾+100c.\overline{abcab}=1000 \cdot \overline{ab} + 100c+\overline{ab}=\underbrace{1001}_{7 \cdot 11\cdot 13} \cdot \overline{ab} + 100c.abcab=1000⋅ab+100c+ab=7⋅11⋅131001⋅ab+100c.
Cum abcab‾ ⋮ 11⇒100⋅c ⋮ 11⇒c=0.\overline{abcab} ~\vdots~ 11 \Rightarrow 100 \cdot c ~\vdots~ 11 \Rightarrow \boxed{c=0}.abcab ⋮ 11⇒100⋅c ⋮ 11⇒c=0. 7⋅11⋅13⋅ab‾ ⋮ 11⋅83⇒7⋅13⋅ab‾ ⋮ 83⏟prim⇒ab‾=83⇒abcab‾=83083.7 \cdot 11 \cdot 13 \cdot \overline{ab} ~\vdots~ 11 \cdot 83 \Rightarrow 7 \cdot 13 \cdot \overline{ab} ~\vdots~ \underbrace{83}_{prim} \Rightarrow \boxed{\overline{ab}=83} \Rightarrow \boxed{\overline{abcab}=83083}.7⋅11⋅13⋅ab ⋮ 11⋅83⇒7⋅13⋅ab ⋮ prim83⇒ab=83⇒abcab=83083.