a.Chứng minh rằng nếu 3x-2y chia hết 17 thì 11x -13y chia hết cho 17 b.Chứng minh rằng nếu 4x+3y chia hết 13 thì 7x +2y chia hết cho 13

a) Ta có:

$\displaystyle  \begin{array}{{>{\displaystyle}l}}
11x\ -\ 13y\\
=\ 17x\ -\ 6x\ -\ 17y\ +\ 4y\\
=\ ( 17x\ -\ 17y) \ -\ ( 6x\ -\ 4y)\\
=\ 17( x\ -\ y) \ -\ 2( 3x\ -\ 2y)
\end{array}$

Vì $\displaystyle 17\ \vdots \ 17\ \Longrightarrow \ 17( x\ -\ y) \ \vdots \ 17$

Mà $\displaystyle 3x\ -\ 2y\ \vdots \ 17\ \Longrightarrow \ -2( 3x\ -\ 2y) \ \vdots \ 17$

Vậy nếu $\displaystyle 3x\ -\ 2y\ \vdots \ 17$ thì $\displaystyle 11x\ -\ 13y\ \vdots \ 17$

b) Ta nhận thấy:

$\displaystyle  \begin{array}{{>{\displaystyle}l}}
13( 3x\ +\ 2y) \ \vdots \ 13\ ( vì\ 13\ \vdots \ 13)\\
8( 4x\ +\ 3y) \ \vdots \ 13\ ( vì\ 4x\ +\ 3y\ \vdots \ 13)\\
\Longrightarrow \ 13( 3x\ +\ 2y) \ -\ 8( 4x\ +\ 3y) \ \vdots \ 13\\
\Longrightarrow \ 7x\ +\ 2y\ \vdots \ 13
\end{array}$

$\displaystyle [ vì\ 13( 3x\ +\ 2y) \ -\ 8( 4x\ +\ 3y) \ =\ 7x\ +\ 2y]$

Vậy nếu $\displaystyle 4x\ +\ 3y\ \vdots \ 13$ thì $\displaystyle 7x\ +\ 2y\ \vdots \ 13$