What is the product of all the factors of the cube of a positive integer n if the product of all the factors of square of n is n3?a)n4b)n6c)n9d)n5e)Cannot be determinedCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Key Data
'n' is a positive integer.
Product of the factors of n² is n³.

If the product of the factors of n² = n³, the only factors of n² are 1, n, and n².
So, we can infer that n does not have any factor other than 1 and itself.
Therefore, n is a prime number.

Factors of n³ if n is a prime number are 1, n, n² and n³.
So, the product of the factors of n³ = 1 × n × n² × n³ = n⁶

Choice B is the correct answer.

Answer

Key Data
'n' is a positive integer.
Product of the factors of n² is n³.

If the product of the factors of n² = n³, the only factors of n² are 1, n, and n².
So, we can infer that n does not have any factor other than 1 and itself.
Therefore, n is a prime number.

Factors of n³ if n is a prime number are 1, n, n² and n³.
So, the product of the factors of n³ = 1 × n × n² × n³ = n⁶

Choice B is the correct answer.

Answer

Understanding the Problem:
To solve this problem, we need to understand the relationship between the factors of a number and the product of those factors.

Factors of a Number:
- Factors of a number are the numbers that can divide the given number without leaving a remainder.
- For example, the factors of 6 are 1, 2, 3, and 6.

Product of Factors:
- The product of all the factors of a number can be calculated by raising the number to the power of half of the total number of factors.
- For example, the product of factors of 6 is 6^(2/2) = 6^1 = 6.

Solution:
- Given that the product of all the factors of n^2 is n^3.
- This means that the product of factors of n^2 is n raised to the power of half of the total number of factors of n^2.
- Therefore, n^3 = n^(number of factors of n^2 / 2).
- Since the number of factors of n^2 is twice the number of factors of n, we can write this as n^3 = n^(2 * number of factors of n / 2).
- Simplifying this, we get n^3 = n^(number of factors of n).
- This implies that the product of all the factors of n is n^3.
- Now, we need to find the product of all the factors of n^3, which will be n^(number of factors of n^3).
- Since the number of factors of n^3 is three times the number of factors of n, the product of all the factors of n^3 will be n^(3 * number of factors of n).
- Therefore, the product of all the factors of n^3 is n^(3 * number of factors of n) = n^(3 * 2) = n^6.
Therefore, the correct answer is option B) n^6.

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