Question

Which of the following numbers are not perfect cubes?

• A. $216$
• B. $128$
• C. $1000$
• D. $100$
• E. $46656$

Answer 1

Answer: (B), (D)

$100$

Step 1. Explanation of correct answer:

Option B. $128$

Numbers that when factorized, its factors can be arranged into triplets of equal factors.

Prime factorization $128$:

$128=\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times 2$

It is clear that $128$ cannot be arranged into triplets of equal factors. So it is not perfect cubes.

Option B is correct option.

Option D. $100$

Prime factorization $100$:

$100=\left(2\times 2\right)\times \left(5\times 5\right)$

It is clear that $100$cannot be arranged into triplets of equal factors. So it is not perfect cubes.

Option D is correct option.

Step 2. Explanation for incorrect options:

Option A. $216$

Numbers that when factorized, its factors can be arranged into triplets of equal factors.

Prime factorization $216$:

$216=\left(2\times 2\times 2\right)\times \left(3\times 3\times 3\right)$

Hence it can be arranged into triplets of equal factors and further can be expressed as follows:

$$\begin{gathered} 216=\left(2\times 3\right)\times \left(2\times 3\right)\times \left(2\times 3\right) \\ =6\times 6\times 6 \\ =6^3 \end{gathered}$$

Option A is incorrect option.

Option C. $1000$

Prime factorization $1000$:

$1000=\left(2\times 2\times 2\right)\times \left(5\times 5\times 5\right)$

Hence it can be arranged into triplets of equal factors and further can be exprss4ed as follows:

$$\begin{gathered} 1000=\left(2\times 5\right)\times \left(2\times 5\right)\times \left(2\times 5\right) \\ =10\times 10\times 10 \\ ={10}^3 \end{gathered}$$

Option C is incorrect option.

Option E. $46656$

Prime factorization $46656$:

$46656=\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times \left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)$

Hence it can be arranged into triplets of equal factors and further can be exprss4ed as follows:

$$\begin{gathered} 46656=\left(2\times 2\times 3\times 3\right)\times \left(2\times 2\times 3\times 3\right)\times \left(2\times 2\times 3\times 3\right) \\ =36\times 36\times 36 \\ ={36}^3 \end{gathered}$$

Option E is incorrect option.

Hence, the correct options are Option B and Option D