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) 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 \(5 \times 2=10\).