Question

Find the smallest number by which the following number must be multiplied to obtain a perfect cube.
A.243
B.256
C.72
D.675
E.100

Answer 1

Hint: In order to find the perfect cube of a number, write the prime factors of the number. If the prime factor occurs in multiple of 3, then the number is a perfect cube else it is not. If the prime factor is not forming a perfect cube then multiply by that prime number to make a number a perfect cube.


Complete step-by-step solution:
243
The prime factors of 243 are${3^3} \times {3^2}$.
Observing its factors it is clear that 243 can become a perfect cube by multiplying it with 3. After multiplication with 3, the number becomes 729 which is a perfect cube.
256
The prime factors of 256 are ${2^3} \times {2^3} \times {2^2}$.
Observing its factors it is clear that 256 can become a perfect cube by multiplying it with 2. After multiplication with 3, the number becomes 512 which is a perfect cube.
72
The prime factors of 256 are${2^3} \times {3^2}$.
Observing its factors it is clear that 72 can become a perfect cube by multiplying it with 3. After multiplication with 3, the number becomes 216 which is a perfect cube.
675
The prime factors of 256 are${3^3} \times {5^2}$.
Observing its factors it is clear that 675 can become a perfect cube by multiplying it with 5. After multiplication with 5, the number becomes 3375 which is a perfect cube.
100
The prime factors of 100 are ${2^2} \times {5^2}$.
Observing its factors it is clear that in 100, there is a deficiency of one 2 and one 5, it can become a perfect cube by multiplying it with 2 and 5. After multiplying it with 2 and 5 or simply 10, the number becomes 1000 which is a perfect cube.


Note: The important step is to decide which number is to be multiplied in an integer to make it a perfect cube. The best way to do this is by doing a prime factorization of a number.
For instance, 216 is a perfect cube. It can be analyzed only when its prime factorization$\left( {{2^3} \times {3^3}} \right)$ is done.