# Subject I. 

For any natural number $\mathrm{n}$, find the remainder of the division of the number $\mathrm{A}=5^{\mathrm{n}+4} \cdot 3^{\mathrm{n}+1}+5^{\mathrm{n}+1} \cdot 3^{\mathrm{n}+3}+3 \cdot 15^{\mathrm{n}}$ by 2013.
## SUBJECT I

For any natural number $\mathrm{n}$, find the remainder of the division of the number $\mathrm{A}=5^{\mathrm{n}+4} \cdot 3^{\mathrm{n}+1}+5^{\mathrm{n}+1} \cdot 3^{\mathrm{n}+3}+3 \cdot 15^{\mathrm{n}}$ by 2013.

## Solution

$$
\mathrm{A}=5^{\mathrm{n}} \cdot 5^{4} \cdot 3^{\mathrm{n}} \cdot 3^{1}+5^{\mathrm{n}} \cdot 5^{1} \cdot 3^{\mathrm{n}} \cdot 3^{3}+3 \cdot 15^{\mathrm{n}}
$$

$\mathrm{A}=15^{\mathrm{n}}\left(5^{4} \cdot 3+5 \cdot 3^{3}+3\right)$

$\mathrm{A}=15^{\mathrm{n}}(1875+135+3)$

$\mathrm{A}=15^{\mathrm{n}} \cdot 2013$, so A is divisible by 2013.

Finalization: the remainder is 0.