Question

Question 46

Fill in the blanks to make the statements true.

The least number by which 72 be multiplied to make it a perfect cube is ___.

Answer 1

3
Resolving 72 into prime factors, we get
$72=2\times 2\times 2\times 3\times 3$
Grouping the factors in triplets of equal factors, we get
$72=(2\times 2\times 2)\times 3\times 3$
We find that 2 occurs as a prime factor of 72 thrice, but 3 occurs as a prime factor only twice. Thus, if we multiply 72 by 3, 3 will also occur as a prime factor thrice and the product will be $2\times 2\times 2\times 3\times 3\times 3$, which is a perfect cube.
Hence, the least number, which should be multiplied by 72 to get a perfect cube is 3.