How to Compute Factorial Using Recursive Odd-Even Number Decomposition

Understanding Factorial Decomposition

What We're Solving: We want to explore how factorials can be broken down into patterns involving odd numbers, even numbers, and powers of 2. This helps us understand the deeper structure hiding within n! and can lead to useful mathematical insights.

The Approach: Think of factorial as having "layers" - instead of just multiplying 1×2×3×4×5×..., we can separate the odd and even factors and see what patterns emerge. This is like sorting ingredients before cooking - it helps us see the recipe more clearly!

Step-by-Step Solution:

Step 1: Understand the basic split Let's start with a small example. For 6!:

• 6! = 1 × 2 × 3 × 4 × 5 × 6
• Odd factors: 1 × 3 × 5
• Even factors: 2 × 4 × 6

Step 2: Analyze the even factors more deeply The even factors can be rewritten by factoring out powers of 2:

• 2 = 2¹ × 1
• 4 = 2² × 1
• 6 = 2¹ × 3

So the even part becomes: 2¹ × 2² × 2¹ × 3 = 2⁴ × 3

Step 3: Develop the general pattern For any factorial n!, we can write:

• n! = (product of odd numbers up to n) × (powers of 2) × (odd factors from even numbers)

Step 4: Create the recursive structure Here's the beautiful insight: if we look at n!, we can express it as:

• n! = (odd part) × 2^k × (factorial-like structure of smaller numbers)

For even n: n! = 1×3×5×...×(n-1) × 2^(n/2) × (n/2)! For odd n: n! = 1×3×5×...×n × 2^((n-1)/2) × ((n-1)/2)!

Step 5: Verify with an example Let's check 6!:

• 6! = 720
• Odd part: 1×3×5 = 15
• Power of 2: 2³ = 8 (since we have three even numbers: 2,4,6)
• Remaining factorial: 3! = 6
• Check: 15 × 8 × 6 = 720 ✗

Wait! Let me reconsider...

Actually: 6! = (1×3×5) × (2×4×6) = 15 × 48 The even part 2×4×6 = 2³ × (1×2×3) = 8 × 6

The Answer: The general recursive formula is:

• For even n: n! = ∏(odd numbers 1 to n-1) × 2^(n/2) × (n/2)!
• For odd n: n! = ∏(odd numbers 1 to n) × 2^((n-1)/2) × ((n-1)/2)!

This captures how factorial "splits" into a product of all odds up to a point, times a power of 2, times a smaller factorial.

Memory Tip: Think of factorial as a "Russian nesting doll" - each factorial contains a smaller factorial inside, wrapped in layers of odd numbers and powers of 2. The pattern repeats at each level, making it truly recursive!

This decomposition is powerful for understanding divisibility properties and appears in advanced topics like combinatorics and number theory. Great question for exploring mathematical structure! 🌟