Total Number of Factors
The total number of factors of a natural number N can be determined from its prime factorization. If N can be expressed as:
N = p1^e1 * p2^e2 * ... * pk^ek
where p1, p2, ..., pk are distinct prime factors and e1, e2, ..., ek are their respective positive integer exponents, then the total number of factors (T) is given by:
T = (e1 + 1)(e2 + 1)...(ek + 1)
Given that T = 45, we need to find the maximum number of distinct prime factors (k).
Factorization of 45
The number 45 can be factored in different ways:
- 45 = 45 = 1 * 45
- 45 = 3 * 15
- 45 = 5 * 9
- 45 = 3 * 3 * 5
We can express 45 in terms of its factors as follows:
- 45 = 9 * 5 = (8 + 1)(4 + 1)
- 45 = 5 * 3 * 3 = (4 + 1)(2 + 1)(2 + 1)
Maximizing Prime Factors
To maximize the number of distinct prime factors (k), we can explore the following combinations leading to 45:
1. k = 3: Possible with (2+1)(2+1)(4+1) = 3 * 3 * 5 (e.g., p1^2 * p2^2 * p3^4).
2. k = 4: Possible with (1+1)(1+1)(1+1)(4+1) = 2 * 2 * 2 * 5 = 40 (not valid).
3. k = 5: Possible with (1+1)(1+1)(1+1)(1+1)(1+1) = 2^5 = 32 (not valid).
Conclusion
The maximum number of distinct prime factors for a natural number N with 45 total factors is 3. This occurs when N has a prime factorization of the form p1^2 * p2^2 * p3^4, providing a balance of exponents to achieve the total of 45 factors.