\(\left(n+1\right)\left(n+2\right)...\left(2n\right)=\frac{1.2.3.....n.\left(n+1\right)\left(n+2\right)...\left(2n\right)}{1.2.3.....n}\)

\(=\frac{1.3.5.....\left(2n-1\right).2.4.6.....\left(2n\right)}{1.2.3.....n}=\frac{1.3.5.....\left(2n-1\right).2^n\left(1.2.3.....n\right)}{1.2.3.....n}\)

\(=1.3.5.....\left(2n-1\right).2^n⋮2^n\).

a, Ta có: 3xy - 5 = x² + 2y

=> 3xy - x² - 2y = 5

=> y.( 3x - 2 ) = 5 + x.x

=> y = \(\frac{5+x^2}{3x-2}\)

=> \(x^2+5⋮3x-2\)( vì y là số nguyên )

=> \(3x^2+15⋮3x-2\)

\(\Rightarrow x\left(3x-2\right)+15+2x⋮3x-2\)

\(\Rightarrow2x+15⋮3x+2\)

\(\Rightarrow6x+45⋮3x+2\)

\(\Rightarrow2.\left(3x+2\right)+41⋮3x+2\)

\(\Rightarrow41⋮3x+2\)

\(\Rightarrow3x+2\in\left\{-41;-1;1;41\right\}\)

\(\Rightarrow3x\in\left\{-43;-3;-1;39\right\}\)

VÌ 3x chia hết cho 3

\(\Rightarrow3x\in\left\{-3;39\right\}\)

\(\Rightarrow x\in\left\{-1;13\right\}\)

+) với x = -1 => y = -6/5 ( loại )

+) với x = 13 => y = 174/37 ( loại )

Vậy không tìm được ( x ; y ) thỏa mãn bài

b,

Xét \(3^{n+2}-2^{n+2}+3^n-2^n=3^n.9-2^n.4+3^n-2^n=3^n.\left(9+1\right)-2^n.\left(4+1\right)=3^n.10-2^n.5\)

\(=3^n.10-2^{n-1}.2.5=3^n.10-2^{n-1}.10=10.\left(3^n-2^{n-1}\right)⋮10\)

\(\Rightarrow3^{n+2}-2^{n+2}+3^n-2^n⋮10\)

Vậy: \(3^{n+2}-2^{n+2}+3^n-2^n⋮10\)

3^{n +} 27 + 3ⁿ + 3 + 2ⁿ + 8 + 2ⁿ + 4 = (3ⁿ + 3ⁿ + 2ⁿ + 2ⁿ ) + 42 chia het cho 6. Suy ra 3ⁿ + 3ⁿ +2ⁿ +2ⁿ chia het cho 6

Vay voi moi so nguyen duong n thi  3ⁿ⁺³+3ⁿ⁺¹+2ⁿ⁺³+2ⁿ⁺² chia hết cho 6

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