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
True.
Solution:
$\because$ a divides b
$\therefore \frac{ b ^3}{ a ^3}=\frac{ b \times b \times b }{ a \times a \times a }=\frac{( ak ) \times( ak ) \times( ak )}{ a \times a \times a }$
$\because$ a divides b
$\therefore b = ak$ fore some k
$\therefore \frac{ b ^3}{ a ^3}=\frac{( ak ) \times( ak ) \times( ak )}{ a \times a \times a }= k ^3$
$\Rightarrow k ^3= b ^3= a ^3\left( k ^3\right)$
$\therefore a^3$ divides $b^3$
Solution:
$\because$ a divides b
$\therefore \frac{ b ^3}{ a ^3}=\frac{ b \times b \times b }{ a \times a \times a }=\frac{( ak ) \times( ak ) \times( ak )}{ a \times a \times a }$
$\because$ a divides b
$\therefore b = ak$ fore some k
$\therefore \frac{ b ^3}{ a ^3}=\frac{( ak ) \times( ak ) \times( ak )}{ a \times a \times a }= k ^3$
$\Rightarrow k ^3= b ^3= a ^3\left( k ^3\right)$
$\therefore a^3$ divides $b^3$
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