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 $$5 \times 2=10$$.