Ways to Express 1000000 as a Product of Three Different Integers

Understanding how to express the number 1000000 as a product of three different integers can provide valuable insights into number theory and algorithmic problem solving. This article aims to explore a methodical approach to determining the number of distinct ways to achieve this using only the prime factors available, while also providing a clear and logical breakdown of the process.

Step-by-Step Guide to Expressing 1000000 as a Product of Three Different Integers

Let's dive into the detailed steps involved:

Prime Factorization

The first step in solving this problem is to obtain the prime factorization of 1000000. We start with the basic expression:

1000000 10^6 2 times; 5^6 2^6 times; 5^6

Total Divisors

Once we have the prime factorization, we can determine the total number of divisors of 1000000 using the formula:

dn (e_1 1)(e_2 1) ... (e_k 1)

Applying this to our prime factorization:

d1000000 (6 1)(6 1) 7 times; 7 49

Counting Triples

The next step involves counting the number of ways to choose three distinct divisors from the 49 available divisors. This is given by the combination formula:

binom{49}{3} frac{49 times; 48 times; 47}{3 times; 2 times; 1} 17296

Validating Factorization

However, not all combinations of the 17296 will yield a valid product of 1000000. To find the valid combinations, we need to consider the assignment of exponents for the prime factors 2 and 5. Given that:

a 2^{x_1} times; 5^{y_1} b 2^{x_2} times; 5^{y_2} c 2^{x_3} times; 5^{y_3}

We need to verify that:

x_1 x_2 x_3 6 y_1 y_2 y_3 6

Counting Non-negative Integer Solutions

The number of non-negative integer solutions to the equations for the exponents can be found using the stars and bars combinatorial method:

Number of solutions binom{6 3 - 1}{3 - 1} binom{8}{2} 28

Total Valid Combinations

Since the choices for x and y are independent, the total number of ways to choose a, b, c such that a times; b times; c 1000000 is:

28 times; 28 784

Conclusion

Thus, the number of ways to express 1000000 as a product of three different integers, ensuring that all orders do not matter, is:

boxed{784}