Final answer:
A positive integer n cannot have more than n factors because each factor is a product of prime numbers with their exponents, and the calculation for the number of factors (adding 1 to exponents and multiplying) will always yield a number less than or equal to n, given that the smallest prime is 2.
Step-by-step explanation:
To prove that a positive integer n cannot have more than n factors, we can use the fundamental property that every positive integer greater than 1 is either prime or can be factored into prime numbers. Let's take the integer n and express it as a product of prime numbers: n = p1^e1 * p2^e2 * ... * pk^ek, where pi are prime factors and ei are their respective exponents.
The number of factors of n can be calculated by adding 1 to each of the exponents and then taking the product of these numbers because each exponent indicates how many times a prime factor can appear in the factors of n (including not appearing at all). Thus, the total number of factors of n is (e1 + 1)*(e2 + 1)*...*(ek + 1).
Now, let us consider the smallest possible value for a prime factor, which is 2. If we were to have n factors, all the exponents would need to be large enough so that when we add 1 and multiply them together, we get n or more. However, as each prime factor is at least 2, and each exponent contributes to a multiplication, the value of n as a product of primes must necessarily be greater than the number of factors calculated.
As a result, the maximum number of factors including 1 and n itself for any positive integer n will always be less than or equal to n. It is impossible to arrange the exponents of the prime factorization in a way that the number of factors exceeds n, as the very first prime factor being 2 already makes the product greater than the sum. Therefore, a positive integer n cannot have more than n factors.