Smallest number multiplication to make perfect cube:

Suppose a number is given, we first have to find its prime factorization. Then, we have to make a pair of three similar numbers. If the number is not a perfect cube, then any one or two numbers will be remaining. We have to multiply those numbers with the same number to make the number a perfect cube.

For example: if the number is 243

Step 1: we need to find the prime factorization of 243

    3 243

    3 81

    3 27

    3 9

    3 3

    1 

Step 2: Making pair of prime factorisation of number 243

243 = 3 * 3 * 3 * 3 * 3

         ———— 

      = 3^3  * 3^2 

This is not a perfect cube. Next, now we will make the number a perfect cube

Step 3: To make the number 243 a perfect cube, the missing numeral is three. So we will multiply 3 to the number 243 to make it a perfect cube.

Step 4: So the perfect cube number is 243 * 3= 729

Smallest number division to make perfect cube:  

Suppose a number is given, we first have to find its prime factorization. Then, we have to make a pair of three similar numbers (since we are dealing with a cube of numbers). If the number is not a perfect cube, then any one or two numbers will be remaining. We have to divtopicIde  those numbers with the same number to eliminate them so that the remaining numbers form a perfect cube.

For example: If the number is 675

Step 1: first we have to find the prime factorization for 675

  3 675

  3 225

  3 75

  5 25

  5 5

1

Step 2: Making pair of prime factorisation of number 675

675 = 3 * 3 * 3 * 5 * 5

         ————

       = 3^3  *5^2.

This is not a perfect cube. So, how can we make the number a perfect cube?

Step 3: To make the number 675 a perfect cube, Here the number 5 is repeated two times instead of three times. So 5* 5 =2 5 

So, we will divtopicIde 25 from the number 675  and get the perfect cube.

Step 4: 675 / 25 = 27 

So the perfect cube number will be 27. 

Mixed Bag Sums: 

Example 1: Is 6655 a perfect cube? If not, find the smallest natural number by which 6655 must be multiplied so that the product is a perfect cube? 

Solution: 

Prime factorisation of 6655

   5 6655

  11 1331

  11 121

  11 11

1

6655= 5 * 11 * 11 * 11

                —————-

        = 5^1 * 11^3

The number 6655 is not a perfect square. 

As the number five is present only once, we need to multiply 5 with 5. Hence, we get 5 * 5 = 25. 

So, we need to multiply 25 with 6655 to make it a perfect cube. 

6655  * 25 = 166375

Example 2: Find the smallest number which should be divtopicIded by 23625 to make it a perfect cube.

Solution: Initially, we have to find the prime factorization of 23625

3 23625

3 7875

3 2625

5 875

5 175

5 35

7 7

        1

23625=  3 * 3 * 3 * 5 * 5 * 5 * 7

          = ————  ————

The number 23625 is not a perfect cube. In prime factorization of 23625, there are pairs of the numbers three and five. Only, the number 7 is left out. So, we will divtopicIde seven from the number to 23625.

So 23625 /7 = 3375

So the number 3375 is a perfect cube.