Question

Aflați cate numere naturale de forma 12ab sunt divizibile simultan cu 5 si cu 3​

Answer 1

Răspuns: $\bf \overline{12ab} \in\{1200, 1230, 1260, 1290,1215,1245, 1275\}\implies 7\: numere$

Explicație pas cu pas:

Salutare!

$\bf \overline{12ab}\:\: \vdots\:\:15$

$\bf a, b - cifre$

$\bf a,b \in\{0,1,2,3,4,5,6,7,8,9\}$

Criteriul de divizibilitate cu 3: "Un număr se divide cu 3 daca si numai daca suma cifrelor numărului respectiv se divide cu 3" ⇒ (1 + 2 + a + b) ∈ {3, 6, 9, 12, 15} ⇒ (3 + a + b) ∈ {3, 6, 9, 12, 15}

Criteriul de divizibilitate cu 5: " Un număr se divide cu 5 daca si numai daca are ultima cifra 0 sau 5" ⇒ b ∈ {0, 5}

• $\bf \text{\bf Daca \underline{b = 0}}$

$\bf \implies 1+2+a+0 = 3\implies a = 3-3\implies \underline{a = 0} \implies \underline{ \overline{12ab} = 1200}$

$\bf \implies 1+2+a+0 = 6\implies a = 6-3\implies \underline{a = 3} \implies \underline{ \overline{12ab} = 1230}$

$\bf \implies 1+2+a+0 = 9\implies a = 9-3\implies \underline{a = 6} \implies \underline{ \overline{12ab} = 1260}$

$\bf \implies 1+2+a+0 = 12\implies a =12-3\implies \underline{a = 9} \implies \underline{ \overline{12ab} = 1290}$

• $\bf \text{\bf Daca \underline{b = 5}}$

$\bf \implies 1+2+a+5 = 9\implies a = 9-8\implies \underline{a = 1} \implies \underline{ \overline{12ab} = 1215}$

$\bf \implies 1+2+a+5 = 12\implies a = 12-8\implies \underline{a = 4} \implies \underline{ \overline{12ab} = 1245}$

$\bf \implies 1+2+a+5 = 15\implies a = 15-8\implies \underline{a = 7} \implies \underline{ \overline{12ab} = 1275}$

$\bf \overline{12ab} \in\{1200, 1230, 1260, 1290,1215,1245, 1275\}\implies 7\: numere$

==pav38==