Question

Fie numerele naturale a, b, c, astfel încât 15a - 14b + 126c = 2016. Arătaţi că a × b este divizibil cu 42.​

Answer 1

$15\,a-14\,b+126\,c=2016$

$15a-14b+126c=2016 \implies \begin{cases} 15a=14b-126c+2016 \\ \\ 14b=15a+126c-2016 \end{cases}$

$15a=14b-126c+2016 = 14\Big(b-9c+144\Big)$

$\implies 15a \; \vdots \; 14 \;\; \mathsf{dar} \; \; (14\; ;15)=1 \implies 14\; \Big| \; a$

$14b=15a+126c-2016=3\Big( 5a+42c-672\Big)$

$\implies 14b \; \vdots \; 3 \;\; \mathsf{dar} \;\; (3\; ;14)=1 \implies 3\; \Big|\;b$

$\left.\begin{aligned} 14\; \Big| \; a \\\\(14\; ; 3)=1 \\ \\ 3\; \Big|\; b \end{aligned} \; \; \right\} \implies 14\cdot 3 \; \Big| \; a \cdot b \iff \boxed{\;\; 42 \; \Big| \; a \cdot b \;\; }$