Question

Write true (T) or false (F) for the following statement:

 If a divides b, then a³ divides b³.

Answer 1

True

\[\because\] a divides b 

 

\[\therefore\]\[\frac{b^3}{a^3} = \frac{b \times b \times b}{a \times a \times a} = \frac{\left( ak \right) \times \left( ak \right) \times \left( ak \right)}{a \times a \times a}\]

\[\because\] a divides b 

\[\therefore\]  b = ak  for some k 

\[\therefore\]

\[\frac{b^3}{a^3} = \frac{\left( ak \right) \times \left( ak \right) \times \left( ak \right)}{a \times a \times a} = k^3 \Rightarrow b^3 = a^3 \left( k^3 \right)\]

\[\therefore\] a³ divides b³