Răspuns: [tex]\pink{\boxed{~\bf \overline{abc} \in \big\{540,541,542,543,544,545,546,547,548,549\big\}~}}[/tex]
Explicație pas cu pas:
[tex]\bf \overline{abc} -\overline{ab} - c = 486[/tex]
[tex]\bf a, b, c - cifre \\[/tex]
[tex]\bf Cifrele~ sunt:0,1,2,3,4,5,6,7,8,9[/tex]
[tex]\bf a, b \neq 0[/tex]
Descompunem in baza zece
[tex]\bf 100a+10b+c -(10a+b) - c = 486[/tex]
[tex]\bf 100a+10b+c -10a-b - c = 486[/tex]
[tex]\bf 90a+9b = 486~~~~~\bigg|:9[/tex]
[tex]\bf 10a+b = 54 \implies \underline{~b = 54 -10a~}[/tex]
Observam ca a ≤ 5 ⇒ a ∈ {1, 2, 3, 4, 5}
dar b = cifra
Din ultimele trei relații ⇒ a = 5 ⇒ b = 54 - 10 · 5 ⇒ b = 54 - 50 ⇒ b = 4
[tex]\implies \red{\bf \overline{ab}= 54}[/tex]
[tex]\bf c -cifra \implies c \in \{0,1,2,3,4,5,6,7,8,9\}[/tex]
[tex]\pink{\boxed{~\bf \overline{abc} \in \big\{540,541,542,543,544,545,546,547,548,549\big\}~}}[/tex]
Verificare:
540 - 54 - 0 = 486 (adevarat)
541 - 54 - 1 = 486 (adevarat)
542 - 54 - 2 = 486 (adevarat) .....
Bafta multa !
