Question

Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube:

1. 243
2. 256
3. 72
4. 675
5. 100

Answer 1

(i) $243$

Prime factors of $243 =3\times3\times3\times3\times3$
Here $3$ does not appear in $3$’s group.
Therefore, $243$ must be multiplied by $3$ to make it a perfect cube.

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(ii) $256$

Prime factors of $256$ = $2\times2\times2\times2\times2\times2\times2\times2$
Here one factor $2$ is required to make a $3$’s group.
Therefore, $256$ must be multiplied by $2$ to make it a perfect cube.

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(iii) $72$

Prime factors of $72$ = $2 \times 2 \times 2 \times 3 \times 3$
Here $3$ does not appear in $3$’s group.
Therefore, $72$ must be multiplied by $3$ to make it a perfect cube.

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(iv) $675$

Prime factors of $675 = 3 \times 3 \times 3\times 5 \times 5$
Here factor $5$ does not appear in $3$’s group.
Therefore $675$ must be multiplied by $5$ to make it a perfect cube.

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(v) $100$

Prime factors of $100 = 2 \times 2 \times 5 \times 5$
Here factor $2$ and $5$ both do not appear in $3$’s group.
Therefore $100$ must be multiplied by $2 \times5$ = $10$ to make it a perfect cube.