Question
Write true $(T)$ or false $(F)$ for the following statement:
If $a$ divides $b$, then $a^3$ divides $b^3$.
If $a$ divides $b$, then $a^3$ divides $b^3$.
✓
Answer
$\because a$ divides $b$
$\therefore\frac{\text{b}^3}{\text{a}^3}=\frac{\text{b}\times\text{b}\times\text{b}}{\text{a}\times\text{a}\times\text{a}}=\frac{\text{(ak)}\times\text{(ak)}\times\text{(ak)}}4{\text{a}\times\text{a}\times\text{a}}$
$\because$ a divides b
$\therefore b = ak$ fore some $k$
$\therefore\frac{\text{b}^3}{\text{a}^3}=\frac{\text{(ak)}\times\text{(ak)}\times\text{(ak)}}{\text{a}\times\text{a}\times\text{a}}
=\text{k}^3$ $\Rightarrow\text{k}^3=\text{b}^3=\text{a}^3(\text{k}^3)$
$\therefore$ $a^3$ divides $b^3$
$\therefore\frac{\text{b}^3}{\text{a}^3}=\frac{\text{b}\times\text{b}\times\text{b}}{\text{a}\times\text{a}\times\text{a}}=\frac{\text{(ak)}\times\text{(ak)}\times\text{(ak)}}4{\text{a}\times\text{a}\times\text{a}}$
$\because$ a divides b
$\therefore b = ak$ fore some $k$
$\therefore\frac{\text{b}^3}{\text{a}^3}=\frac{\text{(ak)}\times\text{(ak)}\times\text{(ak)}}{\text{a}\times\text{a}\times\text{a}}
=\text{k}^3$ $\Rightarrow\text{k}^3=\text{b}^3=\text{a}^3(\text{k}^3)$
$\therefore$ $a^3$ divides $b^3$
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