Question

Which of the following numbers are not perfect cubes: 

1. 216 
2. 128 
3.  1000 
4.  100 
5.  46656

Answer 1

(i) $216$

Prime factors of $216 = 2 × 2 × 2 × 3 × 3 × 3$
Here all factors are in groups of $3$’s (in triplets)
Therefore, $216$ is a perfect cube number.

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(ii) $128$

Prime factors of $128 = 2× 2 × 2 × 2× 2 × 2× 2$
Here one factor $2$ does not appear in a $3$’s group
Therefore, $128$ is not a perfect cube.

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(iii) $1000$

Prime factors of $1000 = 2\times2\times2\times5\times5\times5$
Here all factors appear in $3$’s group.
Therefore, $1000$ is a perfect cube.

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(iv) $100$

Prime factors of $100 = 2 \times 2 \times 5 \times 5$
Here all factors do not appear in $3$’s group.
Therefore, $100$ is not a perfect cube.

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(v) $46656$

Prime factors of $46656 =2 \times2\times2\times2\times2\times2\times3\times3\times3\times3\times3\times3$
Here all factors appear in $3$’s group.
Therefore, $46656$ is a perfect cube.