Units Digit Logic Using Prime Factors

 

❓ Question

Check whether:

$$6^n$$

can end with the digit $0$ for any natural number $n$.

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🖼️ Solution Image

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✍️ Short Solution

This is a prime factorisation + divisibility by 10 question.

A number ending with $0$ must contain both 2 and 5 as prime factors 💯

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🔹 Step 1 — Condition for a Number Ending with 0

For any number to end with digit $0$:

$$\text{Number must be divisible by } 10$$

And,

$$10 = 2 \times 5$$

👉 So the number must contain both 2 and 5 in its prime factors.

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🔹 Step 2 — Prime Factorisation of $6^n$

$$6 = 2 \times 3$$

Therefore,

$$6^n = (2 \times 3)^n$$

👉 Prime factors of $6^n$:

• $2$ ✅
• $3$ ✅

But no factor $5$ ❌

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🔹 Step 3 — Final Conclusion

Since $5$ is not a prime factor of $6^n$,

$$6^n$$

cannot be divisible by $10$.

Hence, it cannot end with digit $0$.

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✅ Final Answer

$$\boxed{6^n \text{ cannot end with digit } 0 \text{ for any natural number } n}$$

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⭐ Key Insight

A number ending with $0$ must always have:

$$2 \times 5$$

as factors.

🧠 Memory Line:

Without factor 5, a number can never end with 0