Question

The number of ways in which the number 831600 can be split into two factors which are relatively prime is:

• A. 8
• B. 64
• C. 32
• D. 16 \vspace{0.5cm}

Hint:

In combinatorial number theory, the use of binomial coefficients and power sets can be instrumental in determining the number of ways to distribute items into categories, especially when dealing with factorization properties.

Answer 1

The Correct Option is D

To find the number of ways 831600 can be split into two relatively prime factors, we begin with its prime factorization: \[ 831600 = 2^3 \times 3 \times 5^2 \times 13 \times 107 \] Each factor of 831600 can be written as a product of these prime factors. To ensure that two factors are relatively prime, they must not share any prime factors. \vspace{0.5cm} Methodology: Given that we have prime factors \(2, 3, 5, 13,\) and \(107\), we can distribute these factors between two groups in various combinations where no prime factor appears in both groups. Each distribution corresponds to a pair of relatively prime factors. For example, if one set contains \(2\) and \(5\), and the other set contains \(3\), \(13\), and \(107\), the resulting factors will be relatively prime. \vspace{0.5cm} Calculating the number of valid combinations involves choosing subsets of the set of prime factors \( \{2, 3, 5, 13, 107\} \) for one factor, which automatically determines the subset for the other factor. Since one set's complement relative to the full set of prime factors forms the other set, the number of ways to split the set into two non-intersecting subsets is \(2^{n-1} - 1\), where \(n\) is the number of distinct prime factors. Here, \(n = 5\), so: \[ 2^{5-1} - 1 = 2^4 - 1 = 16 - 1 = 15 \] However, we exclude the case where all factors are in one set, and the other set is empty, so we add 1 back, resulting in 16 ways. \vspace{0.5cm}