## SUBJECT 2.

a) Determine the natural numbers $n \in \mathbf{N}^{*}$ and $\overline{a b c d}$ written in base 10, knowing that $\overline{a b c d}+\frac{\overline{a b c d}}{6}+\frac{\overline{a b c d}}{6^{2}}+\ldots+\frac{\overline{a b c d}}{6^{\mathrm{n}}}=\frac{6^{\mathrm{n}+1}-1}{5}$.

G.M. $n r .10 / 2014$

b) Show that the number $A=2016^{2015}+2014^{2015}$ has at least 3 prime divisors.
Subject 2.
a) $\overline{a b c d}+\frac{\overline{a b c d}}{6}+\frac{\overline{a b c d}}{6^{2}}+\ldots+\frac{\overline{a b c d}}{6^{n}}=\overline{a b c d} \cdot\left(1+\frac{1}{6}+\frac{1}{6^{2}}+\ldots+\frac{1}{6^{n}}\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots .1 p$

$$
\begin{aligned}
& 1+\frac{1}{6}+\frac{1}{6^{2}}+\ldots+\frac{1}{6^{\mathrm{n}}}=\frac{6^{\mathrm{n}+1}-1}{5 \cdot 6^{\mathrm{n}}} \\
& \overline{\frac{a b c d}{6^{n}}}=1 \Rightarrow \overline{a b c d}=6^{\prime}
\end{aligned}
$$

$\qquad$

![](https://cdn.mathpix.com/cropped/2024_06_07_d87a0d67c55a5fa39cefg-2.jpg?height=66&width=1471&top_left_y=1143&top_left_x=318)
b) $2016^{2015}=(2015+1)^{2015}=M_{2015}+1^{2015}=M_{2015}+1$

$2014^{2015}=(2015-1)^{2015}=M_{2015}-1^{2015}=M_{2015}-1$ $1 p$

$A=M_{2015}+1+M_{2015}-1=M_{2015}=2015 \cdot k, k \in \mathbf{N}^{*}$ $1 p$

$2015=5 \cdot 13 \cdot 31 \Rightarrow A=5 \cdot 13 \cdot 31 \cdot k, k \in \mathbf{N}^{*}$, so A has at least 3 prime divisors.1p