Solution:
[i]81
Prime factors of 81 = 3\times3\times3\times3
Here one factor 3 is not grouped in triplets.
Therefore 81 must be divided by 3 to make it a perfect cube.
[ii] 128
Prime factors of 128 = 2\times2\times2\times2\times2\times2
Here one factor 2 does not appear in a 3’s group.
Therefore, 128 must be divided by 2 to make it a perfect cube.
[iii] 135
Prime factors of 135 = 3\times3\times3\times5
Here one factor 5 does not appear in a triplet.
Therefore, 135 must be divided by 5 to make it a perfect cube.
[iv] 192
Prime factors of 192 = 2\times2\times2\times2\times2\times3
Here one factor 3 does not appear in a triplet.
Therefore, 192 must be divided by 3 to make it a perfect cube.
[v] 704
Prime factors of 704 = 2\times2\times2\times2\times2\times2\times11
Here one factor 11 does not appear in a triplet.
Therefore, 704 must be divided by 11 to make it a perfect cube.
To solve such type of questions i.e quotient will be a perfect cube or perfect square, we always do the prime factorisation of the given number and make the groups in triplet [to calculate perfect cube] and doublet [for perfect square]. Here triplet means group of three same numbers and doublet means group of two same numbers.
Complete step by step solution: We will start here with prime factorisation of our given number:
\[2\]\[18522\]\[3\]\[9261\]\[3\]\[3087\]\[3\]\[1029\]\[7\]\[343\]\[7\]\[49\]\[7\]\[7\]\[1\]
So Prime factorisation of \[18522{\text{ }} = {\text{ }}2 \times 3 \times 3 \times 3 \times 7 \times 7 \times 7\]
Since we have to make quotient a perfect cube, hence we make group of three same number here.
i.e. \[18522{\text{ }} = {\text{ }}2 \times \]\[3 \times 3 \times 3\]\[ \times \]\[7 \times 7 \times 7\]
Clearly it can be seen that 2 is the only factor which is extra here. If we divide our number by 2 here then its quotient can become a perfect cube here.
i.e. \[\dfrac{{18522}}{2} = 9261\] which is a perfect cube of \[21\].
Hence, our desired answer is 2.
Note: Students always remember in prime factorisation you have to use prime numbers. Prime numbers are those which can be divided by one or the number itself. Example \[2,3,5,7 \ldots .\]
Also one is not a prime number. Many students use numbers other than prime and get wrong answers. Secondly, always remember you have to choose the same number to make a triplet because \[{2^3} = 2 \times 2 \times 2\], the cube is always a group of three numbers. We divide only with that number which is not forming a triplet.