How to find if the given number is a perfect cube?
If the prime factors of a number are grouped in triples of equal factors, then that number is called a perfect cube.
Examples to find if the given number is a perfect cube:
1. Find out if the following are perfect cubes.
(i) 250
(ii) 5832
(i) 250
Solution:
Resolving 250 as the product of prime factors
250 = 2 × 5 × 5 × 5
Since 2 does not exist in product of triples.
Therefore, 250 is not a perfect cube.
(ii) 5832
Solution:
Resolving 5832 as the product of prime factors
5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
Since, they are grouped in triples.
Therefore, 5832 is a perfect cube.
2. Find the smallest number by which 1944 must be multiplied so that the product is a perfect cube.
Solution:
Resolving 1944 as the product of prime factors
1944 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3
In order to make 1944 a perfect cube, it must be multiplied by 3.
3. Find the smallest number by which 4394 must be divided so that the quotient is a perfect cube.
Solution:
Resolving 4394 as the product of primes
4394 = 2 × 13 × 13 × 13
In order to make 4394 a perfect cube, it must be divided by 2, so that the quotient is 2197.
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