Question

Determinati valorile naturale ale lui x,pentru care:
a)x+1|x+7
b)x+2|2x+16
c)x-1|3x+11
d)2x+1|4x+11
e)2x-1|6x+12
f)2x+3|5x+15

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Answer 1

Explicație pas cu pas:

a) x+1|x+7

(x + 1) este divizor pentru (x + 7)

x+1 | x+7 <=> x+1 | (x+1) + 6

(x+1) îl divide pe (x+1) => x+1 | 6

<=>:(x+1) este divizor pentru 6

(x+1) ∈ {1, 2, 3, 6}

x+1 = 1 => x = 1 - 1 = 0

x+1 = 2 => x = 2 - 1 = 1

x+1 = 3 => x = 3 - 1 = 2

x+1 = 6 => x = 6 - 1 = 5

=> x ∈ {0, 1, 2, 5}

b) x+2|2x+16

x+2 | 2x+4 + 12 <=> x+2 | 2(x+2) + 12

=> x+2 | 12

(x+2) ∈ {1, 2, 3, 4, 6, 12}

=> x ∈ {0, 1, 2, 4, 10}

c) x-1|3x+11

x-1 | 3x-3 + 14 <=> x-1 | 3(x-1) + 14

=> x-1 | 14

(x-1) ∈ {1, 2, 7, 14}

=> x ∈ {2, 3, 8, 15}

d) 2x+1|4x+11

2x+1 | 4x+2 + 9 <=> 2x+1 | 2(2x+1) + 9

=> 2x+1 | 9

(2x+1) ∈ {1, 3, 9}

2x+1 = 1 => 2x = 0 => x =0

2x+1 = 3 => 2x = 2 => x = 1

2x+1 = 9 => 2x = 8 => x = 4

=> x ∈ {0, 1, 4}

e) 2x-1|6x+12

2x-1 | 6x-3 + 15 <=> 2x-1 | 3(2x-1) + 15

=> 2x-1 | 15

(2x-1) ∈ {1, 3, 5, 15}

=> x ∈ {1, 2, 3, 8}

f) 2x+3|5x+15

2•(5x+15) = 10x + 30

5•(2x+3) = 10x + 15

(10x + 30) - (10x + 15) = 15

=> 2x+3 | 15

(2x+3) ∈ {1, 3, 5, 15}

=> x ∈ {0, 1, 6}