Question

Find whether the following number is a perfect cube or not: 6859

Answer 1

Hint: We solve this problem by using the prime factorisation method. The prime factorization method is nothing but writing the given number in the product of primes. If the product can be written in the form of a cube of some other number then we can say that this number is a perfect cube.

Complete step by step solution:
We are given that the number is 6859
Let us assume that the given number as
\[\Rightarrow n=6859\]
Now, let us use the prime factorization method that is let us divide the given number with all prime numbers starting with ‘2’
Now, let us check which first prime number divides the given number 6859 exactly starting from ‘2’.
So, we can see that the number 19 divides the given number 6859 exactly.
So, we can write the given number as
\[\Rightarrow n=19\times 361\]
Here, we can see that the number 361 is not a prime number.
Now, again by using the prime factorization method for the number 361 we get
\[\Rightarrow n=19\times 19\times 19\]
Here, we can see that all the numbers which are in the product in the above equation are prime numbers and are equal.
So, by rewriting the above equation as a cube of some prime number we get
\[\Rightarrow n={{19}^{3}}\]
Here, we can see that the number 6859 can be written as a cube of 19.
Therefore, we can conclude that the given number 6859 is a perfect cube.

Note: We have a shortcut explanation of the above problem.
In mathematics, we need to remember the squares and cubes of the first 20 natural numbers.
By checking the cubes of all first 20 natural numbers we get that
\[\Rightarrow {{19}^{3}}=6859\]
Here, we can see that the number 6859 can be written as a cube of 19.
Therefore, we can conclude that the given number 6859 is a perfect cube.