Question

Fie x= 35n + 28, unde n este numar natural.
a) Arata ca x este divizibbul cu 7.
b) Afla restul impartirii lui x la 5.

Answer 1

   
[tex]\displaystyle\\ a)\\ x=35^n + 28 = \\ =(7\cdot 5)^n + 7\cdot 4 =\\ = 7^n \cdot 5^n+7\cdot 4=\\ = 7\cdot 7^{n-1}\cdot 5^n+7\cdot 4= \\ = 7\Big( 7^{n-1}\cdot 5^n+4\Big)~\vdots~7\\\\ b)\\ x=35^n + 28 = \\ = 35^n + 25 + 4 \\35^n~\vdots ~5~~\text{si}~~25~\vdots ~5\\ \Longrightarrow~~ 35^n + 25 + 4 = 5k + 4\\ 5k + 4 : 5 = k~rest~\boxed{4} [/tex]