Checking Digit Endings Using Prime Factorization

Discover how the Fundamental Theorem of Arithmetic helps us determine if numbers can end with specific digits!

Prime Factorization & Digit Endings

Learn how to use prime factors to predict
which digits numbers can end with

🎯 The Zero Ending Rule
For a number to end in 0, it must be divisible by both 2 AND 5
10
=
2
×
5
Both prime factors 2 and 5 are required!

🔍 Can 4ⁿ ever end with digit 0?

1
For 4ⁿ to end with 0, it must be divisible by 5
Need: 5 | 4ⁿ
2
Prime factorization must contain 5
Must include factor 5
3
But 4ⁿ = (2²)ⁿ = 2²ⁿ
Only factor 2
4
By uniqueness theorem
∴ 4ⁿ NEVER ends with 0

🔢 Powers of 4 Examples

4¹ = 4
4² = 16
4³ = 64
4⁴ = 256
4⁵ = 1024
Notice: None end with 0!
All end with 4 or 6 (digits that don't require factor 5)

📊 More Examples

❌ Cannot End with 0
8ⁿ = (2³)ⁿ = 2³ⁿ

Only contains prime factor 2
Missing factor 5
Never ends with 0
✅ Can End with 0
10ⁿ = (2×5)ⁿ = 2ⁿ×5ⁿ

Contains both 2 and 5
Has required factors
Always ends with 0

🔍 Can 6ⁿ ever end with digit 0?

1
Find prime factorization of 6
6 = 2 × 3
2
Therefore: 6ⁿ = (2 × 3)ⁿ
6ⁿ = 2ⁿ × 3ⁿ
3
To end with 0, we need factor 5
Missing: factor 5
4
6ⁿ never contains factor 5
∴ 6ⁿ NEVER ends with 0
🎯 Key Insight
The Fundamental Theorem of Arithmetic guarantees that every number has a unique prime factorization. This lets us definitively determine which digits a number can end with by analyzing its prime factors!
📌 To end with 0: Need both 2 AND 5
📌 Missing any required factor = Impossible ending
📌 Unique factorization = Definitive answers
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🧠 Understanding Prime Factorization & Digit Endings
The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorization. This powerful theorem helps us determine which digits numbers can end with by analyzing their prime factors.