Explicație pas cu pas:
22.
[tex]x + 5 \leqslant 17 < = > x \leqslant 17 - 5 = > x \leqslant 12 \\ [/tex]
[tex]x \in \mathbb{N} => x \geqslant 0[/tex]
[tex]A = \{0; 1 ;2 ;3 ;4; 5; 6; 7; 8; 9; 10; 11; 12\} \\ [/tex]
[tex]x = 0 = > 0 + 3 = 3 \\ x = 1 = > 1 + 3 = 4 \\ x = 2 = > 2 + 3 = 5 \\ ... \\ x = 7 = > 7 + 3 = 10 \\ ... \\ x = 12 = > 12 + 3 = 15[/tex]
[tex]5 | (x + 3) => x \in \{2; 7; 12\} \\ [/tex]
[tex]B = \{2; 7; 12\}[/tex]
[tex]{A \cup B } = A[/tex]
[tex]A - B = \{0; 1; 3; 4; 5; 6; 8; 9; 10; 11\} \\[/tex]
23. a)
[tex]S_{1} = 1 + 2 + 3 +...+ 45 = \frac{45 \times 46}{2} \\ = 45 \times 23 = {3}^{2} \times 5 \times 23[/tex]
[tex]S_{2} = 1 + 2 + 3 +...+ 69 = \frac{69 \times 70}{2} \\ = 69 \times 35 = 3 \times 5 \times 7 \times 23[/tex]
[tex]\left(S_{1}; S_{2} \right) =3 \times 5 \times 23 = 345 \\ [/tex]
[tex]\left[S_{1}; S_{2} \right] = {3}^{2} \times 5 \times 7 \times 23 = 7245 \\ [/tex]
23. b)
[tex]S_{1} = 1 + 2 + 3 +...+ 105 = \frac{105 \times 106}{2} \\ = 105 \times 53 = 3 \times 5 \times 7 \times 53[/tex]
[tex]S_{2} = 1 + 2 + 3 +...+ 125 = \frac{125 \times 126}{2} \\ = 125 \times 63 = {3}^{2} \times {5}^{3} \times 7[/tex]
[tex]\left(S_{1}; S_{2} \right) = 3 \times 5 \times 7 = 105 \\ [/tex]
[tex]\left[S_{1}; S_{2} \right] = {3}^{2} \times {5}^{3} \times 7 \times 53 = 417375 \\ [/tex]
