Question

Question 13

Write the correct answer from the given four options.

Which of the following numbers is a perfect cube?

a) 243
b) 216
c) 392
d) 8640

Answer 1

For option (a) We have, 243
Resolving 243 into prime factors, we have
$243=3\times 3\times 3\times 3\times 3$
Grouping the factors in triplets of equal factors, we get
$243=(3\times 3\times 3)\times 3\times 3$
Clearly, in the grouping, the factors in triplets of equal factors, we are left with two factors $3\times 3$
Therefore, 243 is not a perfect cube.

For option (b) We have 216
Resolving 216 into prime factors, we have
$216=2\times 2\times 2\times 3\times 3\times 3$
Grouping the factors in triplets of equal factors, we get
$216=(2\times 2\times 2)\times (3\times 3\times 3)$
Clearly, in grouping, the factors of triplets of equal factors, no factor is left over.
So, 216 is a perfect cube.

For option (c) We have, 392
Resolving 392 into prime factors, we get
$392=2\times 2\times 2\times 7\times 7$
Grouping the factors in triplets of equal factors, we get
$392=(2\times 2\times 2)\times 7\times 7$
Clearly, in grouping, the factors in triplets of equal factors, we are left with two factors $7\times 7$
Therefore, 392 is not a perfect cube.

For option (d), We have, 8640
Resolving into prime factors, we get
$8640=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 5$
Grouping the factors in triplets of equal factors, we get
$8640=(2\times 2\times 2)\times (2\times 2\times 2)\times (3\times 3\times 3)\times 5$
Clearly, in grouping, the factors in triplets of equal factors, we are left with one factor 5.
Therefore, 8640 is not a perfect cube.
After solving, it is clear that option (b) is correct.