Show that 12º cannot end with the digit 0 or 5 for any natural number n

To prove that \(12^n\) cannot end with the digit 0 or 5 for any natural number \(n\), we need to examine the last digit of powers of 12.

### **Step 1: Analyze Powers of 12 Modulo 10**

First, let's consider the last digit of \(12^n\). The last digit of a number is the same as the remainder when the number is divided by 10. This is equivalent to calculating \(12^n \mod 10\).

### **Step 2: Compute the Last Digit of Powers of 12**
Let's calculate the first few powers of 12 and observe the pattern in the last digit:

- \(12^1 = 12\)  
  Last digit = \(12 \mod 10 = 2\)
- \(12^2 = 144\)  
  Last digit = \(144 \mod 10 = 4\)
- \(12^3 = 1728\)  
  Last digit = \(1728 \mod 10 = 8\)
- \(12^4 = 20736\)  
  Last digit = \(20736 \mod 10 = 6\)
- \(12^5 = 248832\)  
  Last digit = \(248832 \mod 10 = 2\)

The sequence of last digits for \(12^n\) is 2, 4, 8, 6, and then it repeats every four terms: 2, 4, 8, 6.

### **Step 3: Identify the Pattern**

The pattern for the last digit of \(12^n\) is as follows:
- \(n \equiv 1 \pmod{4}\) leads to a last digit of 2.
- \(n \equiv 2 \pmod{4}\) leads to a last digit of 4.
- \(n \equiv 3 \pmod{4}\) leads to a last digit of 8.
- \(n \equiv 0 \pmod{4}\) leads to a last digit of 6.

### **Step 4: Conclusion**
- The last digit of \(12^n\) can only be 2, 4, 8, or 6.
- Therefore, \(12^n\) **cannot** end with the digit 0 or 5 for any natural number \(n\).

This completes the proof.

Would you like any further details on this proof?

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Here are 5 related questions:

1. How would you prove that \(2^n\) cannot end with the digit 5 for any natural number \(n\)?
2. What is the last digit of \(7^n\) for different values of \(n\)?
3. Can you generalize a method to find the last digit of any power \(a^n\)?
4. Why is it impossible for \(3^n\) to end in the digit 0 for any \(n\)?
5. How does modular arithmetic help in understanding the properties of last digits in powers?

**Tip:** When dealing with last digits, think in terms of modular arithmetic (mod 10). It simplifies the problem to just examining the remainder when divided by 10.

Explore a mathematical proof showing that the powers of 12, denoted as 12^n, cannot end with the digits 0 or 5 for any natural number n. This analysis uses modular arithmetic to examine the last digits of 12^n and establishes a pattern that excludes the possibility of these specific endings. Suitable for advanced high school students and mathematics enthusiasts.

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