Question

Aratati ca numarul a= 3^2n+2^ •2^n+3^ + 3^2n+1^•2^n+4^ - 3^2n^ • 2^n+5^ este divizibil cu 88
^......^ inseamna la puterea

Answer 1

   

$\displaystyle\bf\\a= 3^{2n+2}\times2^{n+3}+3^{2n+1}\times2^{n+4}-3^{2n}\times2^{n+5}\\\\a= 3^{2n}\cdot3^2\times2^{n}\cdot2^3+3^{2n}\cdot3\times2^{n}\cdot2^4-3^{2n}\times2^{n}\cdot2^5\\\\a= 3^{2n}\times2^{n}\Big(3^2\cdot2^3+ 3\cdot2^4-2^5\Big)\\\\a= 3^{2n}\times2^{n}\Big(9\cdot8+ 3\cdot16-32\Big)\\\\a= 3^{2n}\times2^{n}\Big(72+ 48-32\Big)\\\\\boxed{\bf~a= 3^{2n}\times2^{n}\times88~~\vdots~~88}~~\text{\bf Deoarece un factor = 88}$

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