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Prime factorization is a method used to express a number as the product of its prime factors. Prime numbers are those greater than 1, divisible only by 1 and themselves.
For instance, the number 88,200 can be broken down into its prime factors:

• Twos: 2 raised to the power of 3, written as \(2^3\)
• Threes: 3 raised to the power of 2, written as \(3^2\)
• Fives: 5 raised to the power of 2, written as \(5^2\)
• Sevens: 7 raised to the power of 1, written as \(7^1\)

Thus, the full prime factorization of 88,200 is \(2^3 \cdot 3^2 \cdot 5^2 \cdot 7^1\). Each prime factor occurs a specific number of times, known as its exponent. Prime factorization helps in simplifying mathematical operations like finding the GCD and LCM.
Similarly, 970,200 is factorized as \(2^3 \cdot 3^3 \cdot 5^3 \cdot 7^1\), showing it shares some but not all prime components with the first number.

When we talk about the number of factorizations, we refer to how many different ways a number can be expressed as a product of two of its factors. In this context, the focus is on factorizations where the factors are coprime, meaning their greatest common divisor (GCD) is 1.
To find these coprime pairs, we rely on the modified prime factorization. For 88,200, we first determine the powers of its prime factors: 2, 3, 5, and 7. Assigning these powers to either factor \(a\) or \(b\) ensures they remain coprime. We multiply the number of assignments for each prime factor (e.g., 4 ways for 2, 3 ways for 3, and so forth) resulting in a total e.g., \(4 \cdot 3 \cdot 3 \cdot 2 = 72\) ways for 88,200.
Since order doesn't matter (\(a \cdot b = b \cdot a\)), we divide this total by 2. Therefore, the number of coprime factorizations for:

• 88,200 is 36, and
• 970,200 results in 128 ways, which gives 64 unique pairs.

The concept of numbers being pairwise coprime plays a critical role in finding coprime factorizations. Two numbers are pairwise coprime if their GCD is 1. This indicates that they do not share any prime factors.
When looking to factor a number like 88,200 into pairwise coprime parts, we can assign the prime exponents differently between two factors \(a\) and \(b\) to ensure they don’t share common divisors. By distributing the prime factors in such a way, the product of these factors always equals the number while maintaining their pairwise coprimacy.
For example, assign prime factor 2 raised to the power of 3 \(2^3\) to factor \(a\) and 5 raised to the power of 2 \(5^2\) to \(b\). By assigning different prime bases to \(a\) and \(b\), we ensure \(\text{GCD}(a, b) = 1\). This systematic assignment is essential to compute how many such coprime factor pairs exist for any given number.