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Divisibility by 2

Write true T ...

Question

Write true $(T)$ or False $(F)$ for the following statements:
If $a$ divides $b,$ then $a^{3}$ divides $b^{3}$

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Solution

Step: State true or false

Given statement: If $a$ divides $b,$ then $a^{3}$ divides $b^{3}$

If $a$ divides $b,$
$\Rightarrow \frac{b}{a} = k,$ where $k$ is an integer
$\Rightarrow b = a k$

Now,
$\frac{b^{3}}{a^{3}} = \frac{(a k)^{3}}{a^{3}}$
$= \frac{(k^{3} a^{3})}{a^{3}}$
$= k^{3},$ which is also an integer

( $∵$ Cube of integer is always an integer)

$∴$ If $a$ divides $b,$ then $a^{3}$ divides $b^{3}$ for all values of $a$ and $b .$

Hence, the given statement is true.

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Question 79

In the following question, state whether the given statement is true (T) or false (F).
If a number divides three numbers exactly, it must divide their sum exactly.

Question 80

In the following question, state whether the given statement is true (T) or false (F).
If a number exactly divides the sum of three numbers, it must exactly divide the numbers separately.

(T)

(F)

for the following statements:
If

a

b

are integers such that

a^{2} > b^{2},

a^{3} > b^{3}

(T)

(F)

for the following statements:
If

a^{2}

9,

a^{3}

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Divisibility by 2

Standard VIII Mathematics

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