$\left.\begin{array}{c}\text { Division with remainder } \\ [\text { Arithmetic of remainders (other) })]\end{array}\right]$

Find the remainders of the division

a) $1989 \cdot 1990 \cdot 1991+1992^{2}$ by 7 ;

b) $9^{100}$ by 8 .
The product (sum) of two integers gives the same remainder when divided by $n$ as the product (sum) of their remainders when divided by $n$.

a) The given expression gives the same remainder when divided by 7 as $2 \cdot 3 \cdot 4 + 5^{2} = 49$.

b) The given expression gives the same remainder when divided by 8 as $1^{100} = 1$.

## Answer

a) 0 ; b) 1.

Prove that $n^{3} + 2n$ is divisible by 3 for any natural $n$.

## Solution

$n^{3} + 2n = (n^{3} - n) + 3n$, and the first term is divisible by 3 (see the solution of problem $\left.\underline{30377}\right)$).