ex. 3
M ={ xI x∈Nˣ , x: 12, x ≤ 100}
N ={ xI x∈Nˣ , x: 16, x ≤ 100}
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M= ?, N= ?, M ∩ N
M= { 12, 24, 36. 48, 60. 72. 84, 96.}
M= { 16, 32, 48, 64. 80, 96.}
M ∩ N ={ 48, 96}
ex. 4
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x= ?, ( 12x, 6) = 1
x= 1, 121
x= 5, 125
x= 7, 127
ex. 5
n ∈ IN , (2n + 3) I ( 5n+ 12)
(2n + 3) I ( 5n+ 12) I· 2
(2n + 3) I ( 2n+ 3) I ·5
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(2n - 3) I 10n + 24
(2n - 3) I 10n+ 15
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/ 9 , 2n + 3 ∈ D₉ şi n ∈ IN D₉ = { 1: 3; 9}
2n + 3 = 1 2n + 3 = 3 2n + 3 = 0
2n = 1- 3 2n = 3- 3 2n = 0- 3
2n = -2I:2 2n = 0I:2 2n =-3I: 2
n = -1 ∉ IN n = 0 ∈ IN n = _-3_ ∉ IN 2 Concluzie: n = 0
ex. 6
A = 1+ 3+ 5+ ... + 2 013 , pp=?
A= 1+ 3+ 5+ ... + (2·1 006 + 1)
A = ( 1 006 + 1)²
A= 1 007² , 1 007²= 1 014 049 = pp
ex. 7
r= ?
__________________
n = 1·2·3· ... ·20+ 21
î = 6
( 1·2·3· ... ·20+ 21) : 6 = c ( r) , D = c·î + r, r < î
r= {1, 2, 3, 4, 5 }
ex. 8
N = 2ⁿ⁺¹ ·5ⁿ⁺² + 4 : 9, ∀ n∈ IN
N = 2ⁿ·2¹ ·5ⁿ·5² + 4
N= (2·5)ⁿ·2·25+ 4
N = 10ⁿ·25·2+ 2²
N =10ⁿ·50+ 4 : 9
N = 50 ... 0 + 4 , 5+ 0+ 0 + ...+ 0+ 4= 9 , 9: 9
Obs. Un nr. : 9, dacă suma cifrelor : 9
ex. 9
b = 168
m = 280
p = 224
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nr. max. pachete= ?
168= 2³·3·7
280= 2³·5·7
224= 2⁵·7
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( ... )= 2³·7
