Răspuns:
Explicație pas cu pas:
n ∈ N
c) 5²ⁿ+25ⁿ⁺¹ = (5²)ⁿx1 + 25ⁿx 25 =
= 25ⁿ x 1 + 25ⁿ x 25 =
= 25ⁿ x ( 1+25 ) =
= 25ⁿ x 26 =
= 25ⁿ x 2 x 13 → divizibil cu 13 ( un factor al produsului fiind 13)
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27.
a)
2ⁿ⁺²x3ⁿ + 3ⁿ⁺¹x2ⁿ =
= 2ⁿx2²x3ⁿ + 3ⁿx3¹x2ⁿ =
= 2ⁿx3ⁿx(2²+3) =
= (2x3)ⁿ x ( 4+3) =
= 6ⁿ x 7 → divizibil cu 7
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b)
2²ⁿ⁺¹ x 5ⁿ + 4ⁿ x 5ⁿ⁺¹ =
= 2²ⁿx2¹x5ⁿ + 4ⁿx5ⁿx5¹ =
= (2²)ⁿ x 5ⁿ x 2 + (4x5)ⁿ x 5 =
= 20ⁿx ( 2+5) =
= 20ⁿ x 7 → divizibil cu 7
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c)
2ⁿ⁺³x3ⁿ + 6ⁿ⁺¹ =
= 2ⁿx2³x3ⁿ + 6ⁿx6¹ =
= (2x3)ⁿ x 8 + 6ⁿx6 =
= 6ⁿx (8+6) =
= 6ⁿ x 14 =
= 6ⁿ x 2 x 7 → divizibil cu 7
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d)
25ⁿ⁺¹ x 3²ⁿ⁺² - 5²ⁿx9ⁿ⁺¹ = NU 9ⁿ⁺²
= 25ⁿx25x(3²)ⁿx3² - (5²)ⁿx9ⁿx9¹=
= 25ⁿx9ⁿ x 25x9 - 25ⁿx9ⁿ x 9 =
= 25ⁿx9ⁿ x ( 25x9 - 9) =
= (25x9)ⁿ x 9 x ( 25 - 1) =
= (25x9)ⁿ x 9 x 24 → divizibil cu 24
25ⁿ⁺¹ x 3²ⁿ⁺² - 5²ⁿx9ⁿ⁺²
= 25ⁿx25x(3²)ⁿx3² - (5²)ⁿx9ⁿx9²=
= 25ⁿx9ⁿ x 25x9 - 25ⁿx9ⁿ x 18 =
= 25ⁿx9ⁿ x ( 25x9 - 18) =
= (25x9)ⁿ x 9 x ( 25 - 2) =
= (25x9)ⁿ x 9 x 23 → divizibil cu 23
