A perfect cube is the prime factors of a number that are arranged in triplets of equal factors. Check more about the perfect cubes and How to find if the given number is a perfect cube or not from the Cube and Cube Roots Chapter.
How to Find if the Given Number is a Perfect Cube?
To Find if the Given Number is a Perfect Cube, we need to find the prime factors of that number. If the prime factors are grouped together with the triplets, then the number is considered as the perfect cube. If not, the number is not a perfect cube. If you see the prime factor is a single factor or it is a double factor, then that particular number is not treated as a perfect cube.
Let’s find whether the given numbers are perfect cubes or not.
1. Find the below Numbers are Perfect Cubes?
(i) 18 (ii) 27,000 (iii) 120
Answer:
(i) 18
First, find the prime factors of the given number.
The prime factors of the given number 18 are 2, 3, 3
Therefore, 18 = 2 × 3 × 3
The prime factors 2 and 3 are single and double factors. There is no triple factor available in the prime factors of the given number. So, the given number is not a perfect cube.
The given number 18 is not a perfect cube.
(ii) 27,000
First, find the prime factors of the given number.
The prime factors of the given number 27,000 are 30, 30, 30
Therefore, 27,000 = 30 × 30 × 30
The prime factor 30 is grouped in triples. There is no other prime factor is left as single and double factors. So, the given number is a perfect cube.
The given number 27,000 is a perfect cube.
(i) 120
First, find the prime factors of the given number.
The prime factors of the given number 120 are 2, 2, 2, 3, 5
Therefore, 120 = 2 × 2 × 2 × 3 × 5
The prime factors 3 and 5 are single factors. There is one prime factor grouped in triples. As the single prime factors available in the product of prime factors of the given number, it is is not a perfect cube.
The given number 120 is not a perfect cube.
Smallest Multiple that is a Perfect Cube
The smallest multiple that make a number perfect cube can be calculated by following the step by step method product of prime factors of a number. A clear explanation is given To Find if the Given Number is a Perfect Cube and what is the smallest multiple that needs to make it a perfect cube. Check the below examples for a better understanding.
2. Find the smallest number by which 1372 must be multiplied so that the product is a perfect cube?
To find the smallest number by which 1372 must be multiplied so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 1372 are 7, 7, 7, 2, 2
1372 = 7 × 7 × 7 × 2 × 2
1372 = 7³ × 2 × 2
From above, by multiplying 2 the number 1372 becomes the perfect cube.
Therefore, the smallest number is 2.
3. Find the smallest number by which 2000 must be multiplied so that the product is a perfect cube?
To find the smallest number by which 2000 must be multiplied so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 2000 are 5, 5, 5, 4, 4
2000 = 5 × 5 × 5 × 4 × 4
2000 = 5³ × 4 × 4
From above, by multiplying 4 the number 2000 becomes the perfect cube.
Therefore, the smallest number is 4.