Question

Aflati toate nr. de forma abc stiind ca impartind nr abc la bc obtinem catul 5 si restul 4​

Answer 1

 

$\displaystyle\bf\\\overline{abc}:\overline{bc}=5~rest~4\\\\\overline{abc}=\overline{a00}+\overline{bc}\\\\\overline{abc}:\overline{bc}=(\overline{a00}+\overline{bc}):\overline{bc}=5~rest~4$

.

$\displaystyle\bf\\Desfacem~paranteza.\\\\(\overline{a00}+\overline{bc}):\overline{bc}=\overline{a00}:\overline{bc}+\overline{bc}:\overline{bc}=5~rest~4\\\\\overline{bc}:\overline{bc}=1\\\\\implies~\overline{a00}:\overline{bc}=4~rest~4$

.

$\displaystyle\bf\\a~poate~fi~orice~cifra~in~afara~de~0.\\\\a=1\implies 100:\overline{bc}=4~rest~4\implies (100-4):\overline{bc}=4\\\implies\overline{bc}=(100-4):4=96:4=24\implies~\boxed{\bf~a=1;~b=2;~c=4}\\\\a=2\implies 200:\overline{bc}=4~rest~4\implies (200-4):\overline{bc}=4\\\implies\overline{bc}=(200-4):4=196:4=49\implies~\boxed{\bf~a=2;~b=4;~c=9}$

.

$\displaystyle\bf\\a=3\implies 300:\overline{bc}=4~rest~4\implies (300-4):\overline{bc}=4\\\implies\overline{bc}=(300-4):4=296:4=74\implies~\boxed{\bf~a=3;~b=7;~c=4}\\\\a=4\implies 400:\overline{bc}=4~rest~4\implies (400-4):\overline{bc}=4\\\implies\overline{bc}=(400-4):4=396:4=99\implies~\boxed{\bf~a=4;~b=9;~c=9}$

Nu mai putem continua cu valor mai mari decat 4 pentru a

deoarece  vom obtine valori mai mari decat 99 (3 cifre) pentru bc, dar bc este un numar de 2 cifre.

Problema are 4 solutii:

abc ∈ {124; 249; 374; 499}