2.Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.
(i) 243

(ii) 256

(iii) 72

(iv) 675

(v) 100

(i) We have the prime factors of 243 as:

\(\qquad \begin{array}{c|lcr} 3 & 243\\ \hline 3 & 81\\ \hline 3 & 27\\ \hline 3 & 9\\ \hline 3 & 3\\ \hline & 1 \end{array} \)

\(243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^3 \times 3 \times 3\)

We can observe clearly that, number 3 is required to make \(3 \times 3\) a group of three. Thus, the required smallest number to be multiplied is 3.

(ii) We have the prime factors of 256 as:

\(\qquad \begin{array}{c|lcr} 2 & 256\\ \hline 2 & 128\\ \hline 2 & 64\\ \hline 2 & 32\\ \hline 2 & 16\\ \hline 2 & 8\\ \hline 2 & 4\\ \hline 2 & 2\\ \hline & 1\\ \end{array} \)

\(256 =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^3 \times 2^3 \times 2 \times 2\)

We can observe clearly that, number 2 is required to make \(2 \times 2 \) a group of three.Thus, the required smallest number to be multiplied is 2.

(iii) We have the prime factors of 72 as:

\(\qquad \begin{array}{c|lcr} 2& 72\\ \hline 2 & 36\\ \hline 2 & 18\\ \hline 3 & 9\\ \hline 3 & 3\\ \hline & 1 \end{array} \)

\(72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3 \times 3\)

We can observe clearly that, number 3 is required to make \(3 \times 3\) a group of three.Thus, the required smallest number to be multiplied is 3.

(iv) We have the prime factors of 675 as:

\(\qquad \begin{array}{c|lcr} 3 & 675\\ \hline 3 & 225\\ \hline 3 & 75\\ \hline 5 & 25\\ \hline 5 & 5\\ \hline & 1 \end{array} \)

\(675 = 3 \times 3 \times 3 \times 5 \times 5 = 3^3 \times 5 \times 5\)

We can observe clearly that, number 5 is required to make \(5 \times 5\) a group of three.Thus, the required smallest number to be multiplied is 5.

(v) We have the prime factors of 100 as:

\(\qquad \begin{array}{c|lcr} 2 & 100\\ \hline 2 & 50\\ \hline 5 & 25\\ \hline 5 & 5\\ \hline & 1 \end{array} \)

\(243 = 2 \times 2 \times 5 \times 5 \)

We can observe clearly that, number 2 and 5 are required to make \(2 \times 2\) and \(5 \times 5\) a group of three.Thus, the required smallest number to be multiplied is \(5 \times 2=10\).