For all integers a, b and c, if 𝑎|𝑏 and 𝑏|𝑐, then prove that 𝑎𝑏 2 |𝑐 3 .

For all integers a, b, and c if $a | b$ and $b | c$ , then prove that $a b^{2} | c^{3}$

By using definition of divisibility as if $a|b$ can be written in the equation as $b=a m, \quad m \in Z$ and $b | c$ then $c=b{n}$ where, m, n are integer. $m, n \in z$

$\therefore$ By using definition of divisibility

$\therefore$ we have to prove $a b^{2} | c^{3}$

$\Rightarrow c^{3}=(b n)^{3} \quad \because c=b n\quad as \quad b | c$

$c^{3}=b b^{2} n^{3}$ as $a \mid b$

$c^{3}=(a m) b^{2} n^{3}\quad \because b=a m$

$c^{3}=a b^{2}(m)\left(n^{3}\right)$ where $m, n^{3} \in z$

$k=\left(\mathrm{mn}^{3}\right)$

which means $a b^{2}| c^{3}$ by definition of divisibility.