Question

each prime factor appears _______ time in its cube​

Answer 1

Answer:

4 times in cube

Step-by-step explanation:

Example any cube number is 6 cube =216

Now finding prime factors of 216

216 =2×108

2×2×54

2×2×2×27

2×2×2×3×9

2×2×2×3×3×3

2cube= 3 cube

2 times, 3 times

similarly for remaining cube number also

same process

I hope this answer will help u

Answer 2

Therefore each prime factor appears 3 times or a multiple of 3 based on the number of times the prime number gets repeated in its cube.

Given:

A number to be cubed.

To Find:

Each prime factor appears how many times in its cube.

Solution:

The given question can be easily solved by taking various cases as shown below.

Let the number be 'N'.

Case-1: The given number be a prime number:

Let the number 'N' be '3'.

Then N³ = 3³

Hence in this case each prime factor appears '3' times in its cube.

Case-2: The given number is a product of non-repeating prime numbers:

Let the number 'N' be '6'.

Then N³ = 6³ = ( 2 × 3 )³ = 2³ × 3³

Hence in this case also each prime factor appears '3' times in its cube.

Case-3: The given number is a product of repeating prime numbers:

Let the number 'N' be '12'.

Then N³ = 12³ = ( 2 × 2 × 3 )³ = 2³ × 2³ × 3³ = 2⁶ × 3³

Hence, in this case, the repeating prime factors appear twice the number of times repeating [ i.e., if the prime number repeats twice in the number that prime number appears '6' times in the cube and if the prime number repeats thrice in the number that prime number appears '9' times in the cube. ]

To generalize the solution, if a number 'N' has its prime number 'p' repeating 'a' times then it is found to appear '3a' times in its cube.

Therefore each prime factor appears 3 times or a multiple of 3 based on the number of times the prime number gets repeated in its cube.

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