Question

Ma puteti ajuta la matematica dau coronita

Answer 1

$a) 2^3\cdot 5^2-2^2\cdot 5=2^2\cdot 5(2\cdot 5-1)=2^2\cdot 5\cdot (10-1) = 2^2\cdot 5\cdot 9~\vdots~ 9\\b)3^8\cdot 5^7 +3^6\cdot 5^8=3^6\cdot 5^7\cdot (3^2+5)=3^6\cdot 5^6 \cdot 14~ \vdots~ 7\\c) 2^{n+1}\cdot 3^n+2^n\cdot 3^{n+1} = 2^n\cdot 3^n\cdot 2+2^n\cdot 3^n\cdot 3=5\cdot 2^n\cdot 3^n~ \vdots~ 5\\\\9. 7a+b=98\\b= 98-7a\\b= 7(14-a)\\\text{Deoarece membrul drept este multiplu de 7, atunci si membrul stang}\\ \text{trebuie sa fie.Deci }\exists k\in \mathbb{N}^*,\text{astfel incat }b=7k$

$7k=7(14-a)\\k=14-a\\k+a=14\\\text{De aici obtinem solutiile }(a,k)\in \{(0,14),(1,13),\ldots, (14,0)\}\text{,de unde }\\(a,b)\in \{(0,98),(1,92),\ldots , (13,1) , (14,0)}$