Question

What is the smallest number by which \[1600\] must be divided so that the quotient is a perfect cube?

Answer 1

Hint: To solve this question, we will start with factorising the given number \[1600,\] where we will get some factors of the number, then we will make group of three (because as it is given that it needs to be perfect cube), so, after arranging the factors into group of three, the remaining number will be our required answer.

Complete step-by-step answer:
We have been given a number, i.e., \[1600,\] we need to find the smallest number by which \[1600\] must be divided so that the quotient is a perfect cube.
The given number is \[1600,\] so we will start with factorizing the number \[1600.\]
On factorization the number, we get
\[1600{\text{ }} = {\text{ }}2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5\]
So, we get the factors of \[1600,\] now we need to group the numbers in a group of three since it is given in the question that it has to be a perfect cube.
We can see above that the number \[5\] does not form a triplet, because it only contains two \[5\prime s.\]
Hence, the number, \[5 \times 5 = 25,\] i.e., \[25\] has to be divided so that the quotient becomes a perfect cube.
Thus, the smallest number by which \[1600\] must be divided so that the quotient is a perfect cube is \[25.\]

Note: In the question, we were asked about the perfect cube, so, a perfect cube is an integer that is equal to some other integer which is raised to the third power. We refer to the perfect cube by raising a number to the third power as cubing the number.
For example, \[8\] is a perfect cube because of \[2,\] because, \[2 \times 2 \times 2 = 8.\]