Question

If a divides b, then ${{a}^{3}}$ divides ${{b}^{3}}$.
[a] True.
[b] False.
[c] Ambiguous.
[d] Insufficient information.

Answer 1

Hint: In mathematics, in order to prove that a statement is correct, we have to come up with formal proof, and in order to prove that the statement is incorrect, we have to come up with a counterexample. We say an integer “a” divides another integer “b” if and only if there exists an integer k such that b = ak. Mathematically, we write a|b. Using the above definition try proving that the above statement holds true or come up with a counterexample to disprove the statement.

Complete step-by-step answer:
We are given that a divides b.
Hence from the definition, we have, there exists an integer k such that a=bk.
Cubing both sides, we get
${{a}^{3}}={{k}^{3}}{{b}^{3}}$.
Now, since k is an integer, ${{k}^{3}}$ is also an integer.
So let ${{k}^{3}}=z$.
So we have ${{a}^{3}}=z{{b}^{3}}$, where z is an integer. Hence, ${{a}^{3}}$ divides ${{b}^{3}}$.
Hence the statement is true,
Hence option [a] is correct.

Note: [1] a|b if and only if gcd(a,b) = a.
Also, we know that $\gcd \left( {{a}^{n}},{{b}^{n}} \right)={{\left( \gcd \left( a,b \right) \right)}^{n}}$
Put n = 3, we get
$\gcd \left( {{a}^{3}},{{b}^{3}} \right)={{\left( \gcd \left( a,b \right) \right)}^{3}}$
Now since a|b, gcd(a,b) =a.
Hence we have $\gcd \left( {{a}^{3}},{{b}^{3}} \right)={{\left( a \right)}^{3}}={{a}^{3}}$
Hence, we have ${{a}^{3}}|{{b}^{3}}$.
Hence the statement is true,
Hence option [a] is correct.