Question

1.Cate perechi (a,b) de numere naturale nenule,cu a<b,au proprietatea ca (a,b)=15 si [a,b]=360?

A. 4
B. 2
C. 8
D. 6

2.Masura unghiului format din acele din ceasornic care arata ora 12 si jumatate este egala cu:

A. 180°
B. 165°
C. 150°
D. 90°​

Answer 1

$\boxed{\bf \bigg(a\, ; b \bigg) \cdot \bigg[a\, ; b\bigg] = a \cdot b}$

$\bf \bigg(a\, ; b \bigg) = 15$

$\bf \bigg[ \: a\, ; b \bigg]=360$

$\bf \implies a \cdot b = 15 \cdot 360 =5400$

$\bf \bigg(a\, ; b \bigg) = 15 \implies \Bigg\{ a=15k \\\bf \bigg(a\, ; b \bigg) = 15 \implies \Bigg \{ b = 15 p$

$\mathbf{(k\, ; p )=1 } \mathsf{\; -prime \; intre\; ele}$

$\bf a \cdot b = 5400 \iff 15k \cdot 15p = 5400$

$\bf 15^2 \cdot k \cdot p = 5400\; \bigg| :15^2 \iff k \cdot p = 24$

$\bf D_{24}= \{1;2;3;4;6;8;12;24\} \implies (k\, ; p) \in D_{24}$

$\bf dar \; \bigg(k\, ;p\bigg)=1 \implies \bigg(k\, ; p \bigg) \in \bigg\{ \bigg(1\, ;24\bigg); \bigg(3\, ;8\bigg);\bigg(8\, ;3\bigg);\bigg(24\, ;1\bigg)\bigg\}$

$\bf \iff \Bigg\{ a=15k=15 \rightarrow b = 15p =15 \cdot 24=360$

$\bf \iff \Bigg\{ a=15k=15 \cdot 3=45 \rightarrow b=15p=15 \cdot 8 = 120$

$\bf \iff \Bigg\{ a=15k=15 \cdot 8 = 120 \rightarrow b = 15p = 15 \cdot 3 = 45$

$\bf \iff \Bigg\{ a=15k = 15 \cdot 24=360 \rightarrow b=15p=15$

$\bf Dar : \; a < b \implies \boxed{\bf \bigg(a\, ;b\bigg)\in \Bigg\{ \bigg(360\, ; 15\bigg);\bigg(120\, ;45\bigg) \Bigg\} }$

$\mathsf{Avem \; doua \; solutii\; \implies \boxed{ \bf Raspunsul \; este \; B }}$

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$\mathsf{Privind \; desenul \; atasat \; observam\; ca \; acul\; de \; la\; minutul\; 30\; cade \; pe \; ora \; 6\; (18) : }$        

• $\bf 12 \rightarrow 13 \; = 1\, h$

• $\bf 13 \rightarrow 14 = 1\, h$

• $\bf 14 \rightarrow 15 = 1\, h$

• $\bf 15 \rightarrow 16 = 1\, h$

• $\bf 16 \rightarrow 17 = 1\, h$

• $\bf 17 \rightarrow 18 = 1\, h$

$\bf \iff 12 \rightarrow 18 = 6h$

$\bf ceasul=cerc \implies ceasul = 360^{\circ}$

$\mathbf{ceasul = 12\, h \implies 360^{\circ} : 12\, h =30^{\circ} }$

$\bf \implies 6\, h = 6 \cdot 30^{\circ} \implies \boxed{\bf \angle= 180^{\circ} (A)}$

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