la 4) a):
[tex] (a+b)^n = a^n + (C_n^1)\cdot a^{n-1}b +\\+(C_n^2)\cdot a^{n-2}b^2 +...+\\+(C_{n-1}^n)\cdot ab^{n-1}+b^n = \\ \\ = a\cdot \Big(a^{n-1} + (C_n^1)\cdot a^{n-2}b +\\+ (C_n^2)\cdot a^{n-3}b^2 +...+\\+(C_{n-1}^n)\cdot b^{n-1}\Big) + b^n\\ 5:4 = 1 \text{ rest } 1 \Rightarrow 5 = 1\cdot 4 + 1\Big|^n \Rightarrow \\ \\ \Rightarrow 5^n = (4+1)^n = 4\cdot(c) +1^n = (c)\cdot 4 + 1 \Rightarrow \\ \\ \Rightarrow 5^n:4 = (c) \text{ rest } 1 \Big|+3 \Rightarrow \\ \\ \Rightarrow (5^n+3):4 = (c) \text{ rest } 4 \Rightarrow \\ \\ \Ruggtarriw (5^n+3):4 = (c+1) \text{ rest } 0 \Rightarrow 5^n+3~\vdots~4 [/tex]
