Remainder

A is an arbitrary integer that gives remainder 1 in the division with 6. B is an arbitrary integer that gives remainder 2 the division by. What makes remainder in division by 3 product of numbers A x B ?

Correct result:

x =  2

Solution:

$$\begin{gathered} A=1069 \\ B=326 \\ C=A\cdot B=1069\cdot 326=348494 \\ x=C-3\cdot \lfloor C\mathrm{/}3\rfloor =348494-3\cdot \lfloor 348494\mathrm{/}3\rfloor =2 \\ A=6n+1 \\ B=3m+2 \\ C=A\cdot B=(6n+1)(3m+2)=18mn+3m+12n+2=3(6mn+m+4n)+2 \end{gathered}$$

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