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 $gcd(a^n,b^n)={(gcd(a,b))}^n$
Put n = 3, we get
$gcd(a^3,b^3)={(gcd(a,b))}^3$
Now since a|b, gcd(a,b) =a.
Hence we have $gcd(a^3,b^3)={\left(a\right)}^3=a^3$
Hence, we have $a^3|b^3$.
Hence the statement is true,
Hence option [a] is correct.