Question 69

State whether the following statement is true or false:

Let x and y be natural numbers. If x divides y, then $x^{3}$ divides $y^{3}$

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Solution

True
If x divides y, then $\frac{y}{x}$ is a natural number
$\Rightarrow {(\frac{y}{x})}^{3}$ is also a natural number
$\Rightarrow \frac{y^{3}}{x^{3}}$ is a natural number
$\Rightarrow x^{3}$ divides $y^{3}$

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Question 69 State whether the following statement is true or false: Let x and y be natural numbers. If x divides y, then x3 divides y3

True If x divides y, then yx is a natural number ⇒(yx)3 is also a natural number ⇒y3x3 is a natural number ⇒x3 divides y3

Question 69

State whether the following statement is true or false:

Let x and y be natural numbers. If x divides y, then $x^{3}$ divides $y^{3}$

True
If x divides y, then $\frac{y}{x}$ is a natural number
$\Rightarrow {(\frac{y}{x})}^{3}$ is also a natural number
$\Rightarrow \frac{y^{3}}{x^{3}}$ is a natural number
$\Rightarrow x^{3}$ divides $y^{3}$