Question

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

• A. $1323$
• B. $7803$
• C. $35721$
• D. $2560$

Answer 1

The correct option is A $1323$

$\left(i\right)1323$

We see that $1323=3\times 3\times 3\times 7\times 7$

So $7$ does not occur in triplets,

$\therefore 1323$ is not a perfect cube.

So, we multiply $7$ to make triplet

So, our number becomes $1323\times 7=3\times 3\times 3\times 7\times 7\times 7=9621$

Hence $7$ is the smallest number to be multiplied with $1323$ to get a perfect cube number $9621$

Cube root of $9621=3\times 7=21$

$\left(ii\right)7803$

We see that $7803=3\times 3\times 3\times 17\times 17$

So $17$ does not occur in triplets,

$\therefore 7803$ is not a perfect cube.

So, we multiply $17$ to make triplet

So, our number becomes $7803\times 17=3\times 3\times 3\times 17\times 17\times 17=132,651‬$

Hence $17$ is the smallest number to be multiplied with $7803$ to get a perfect cube number $132,651‬$

Cube root of $132,651‬=3\times 17=51$

$\left(iii\right)35721$

We see that $35721=3\times 3\times 3\times 3\times 3\times 3\times 7\times 7$

$35721=9\times 9\times 9\times 7\times 7$

So $7$ does not occur in triplets,

$\therefore 35721$ is not a perfect cube.

So, we multiply $7$ to make triplet

So, our number becomes $35721\times 7=9\times 9\times 9\times 7\times 7\times 7=250,047‬‬$

Hence $7$ is the smallest number to be multiplied with $35721‬‬$ to get a perfect cube number $250,047‬‬$

Cube root of $250,047‬=9\times 7=63$

$\left(iv\right)2560$

We see that $2560=5\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$

$2560=5\times 8\times 8\times 8$

So $5$ does not occur in triplets,

$\therefore 2560$ is not a perfect cube.

So, we multiply $25$ to make triplet

So, our number becomes $2560\times 25=8\times 8\times 8\times 5\times 25=64,000$

or $2560\times 25=8\times 8\times 8\times 5\times 5\times 5=64,000$

Hence $25$ is the smallest number to be multiplied with $2560$ to get a perfect cube number $64,000$

Cube root of $64,000‬=8\times 5=40$