CMR: A = 11¹⁵ + 11¹⁴ - 11¹³ ⋮ 131

          A =  11¹³.(11² + 11 - 1)

          A = 11¹³.131 

          vì 131 ⋮ 131 ⇒ A = 11¹³.131 ⋮ 131 (đpcm)

Ta có: 9 > 8 => 9¹⁶ > 8¹⁶ = (2³)¹⁶ = 2⁴⁸

Do đó: 9¹⁶ > 2⁴⁸

Mặt khác: 15¹¹ < 16¹¹ = (2⁴)¹¹ = 2⁴⁴

Nên 15¹¹ < 2⁴⁴

Do t/c bắc cầu: 9¹⁶ > 2⁴⁸ > 2⁴⁴ > 15¹¹

=> 9¹⁶ > 15¹¹

--> đpcm

Ta có: \(2^{17}+2^{14}\)

\(=2^{14}\left(2^3+1\right)=2^{14}\times9⋮9\)

\(15^3-25^2\)

\(=3^3.5^3-5^4\)

\(=5^3\left(27-5\right)=5^3.2.11⋮11\)

\(2^{17}+2^{14}=2^{14}\left(2^3+1\right)=2^{14}\cdot9\Rightarrow2^{17}+2^{14}⋮9\)

\(15^3-25^2=3^3\cdot5^3-5^4=5^3\left(3^3-5\right)=5^3\cdot22=5^3\cdot11\cdot2\Rightarrow15^3-25^2⋮11\)

\(15^3-25^2⋮11\)

\(3^3.5^3-5^4\)

\(=3^3.5^3-5^3.5\)

\(5^3\left(3^3-5\right)\)

\(=5^3.\left(22\right)⋮11\)

=> đpcm

a) Ta có :

15³ - 25² = ( 5 . 3 )³ - ( 5² )² = 5³ . 3³ - 5^2.2

= 5³ . 27 - 5⁴ = 5³ . 27 - 5³ . 5 = 5³ . ( 27 - 5 ) = 5³ . 22

Vì 22 chia hết cho 11 

=> 5³ . 22 chia hết cho 11

=> 15³ - 25² chia hết cho 11

Ta có : 15³ - 25²  = (5 . 3 )³ -(5²)² = 5³ . 3³ - 5^2.2

= 5³ . 27 - 5⁴ = 5³ . 27 - 5³ . 5 = 5³ . ( 27 - 5 ) = 5³ . 22

Vì 22 chia hết cho 11 => 5³ . 22 \(⋮\)11 => (15³ - 25² ) chia hết cho 11

Vậy (15³-25²) chia hết cho 11

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tHAM KHẢO TRONG ĐÂY NÈ

đặt x=11...11(n+1 chữ số 1)

a=x+4;b=x+8

ab+4=(x+4)(x+8)+4

        =x^2+12x+32+4

         =(x+6)^2 cp

\(A=\frac{15}{11}+\frac{5}{12}+\frac{11}{13}+\frac{9}{14}\)

\(\Rightarrow A=\left(1+\frac{4}{11}\right)+\left(1-\frac{7}{12}\right)+\left(1-\frac{2}{13}\right)+\left(1-\frac{5}{14}\right)\)

\(\Rightarrow A=\left(1+1+1+1\right)-\left(\frac{7}{12}-\frac{4}{11}+\frac{2}{13}+\frac{5}{14}\right)\)

\(\Rightarrow A=4-\left(\frac{7}{12}-\frac{4}{11}+\frac{2}{13}+\frac{5}{14}\right)< 4\)