The remainder when `6^(6^6^6^6^(..oo "times"))` is divided by 10

Options

A. 3
B. 6
C. 0
D. can't be determined

Answer

To find the remainder when \( 6^{6^{6^{6^{\cdots}}}} \) (with infinite exponentiation) is divided by 10, we can follow these steps:

### Step 1: Understand the Pattern of Powers of 6 Modulo 10
First, we need to find the pattern of \( 6^n \mod 10 \) for various values of \( n \):
- \( 6^1 = 6 \) (remainder 6 when divided by 10)
- \( 6^2 = 36 \) (remainder 6 when divided by 10)
- \( 6^3 = 216 \) (remainder 6 when divided by 10)
- \( 6^4 = 1296 \) (remainder 6 when divided by 10)

From this, we can see that for any positive integer \( n \), \( 6^n \mod 10 = 6 \).

### Step 2: Apply the Pattern to the Infinite Exponentiation
Since the expression \( 6^{6^{6^{6^{\cdots}}}} \) is an infinite exponentiation, we can denote it as \( x = 6^{6^{6^{\cdots}}} \).

### Step 3: Determine the Remainder
From our findings in Step 1, we know that no matter how high the power of 6 is, the remainder when divided by 10 is always 6. Therefore, \( 6^{6^{6^{6^{\cdots}}}} \mod 10 = 6 \).

### Conclusion
Thus, the remainder when \( 6^{6^{6^{6^{\cdots}}}} \) is divided by 10 is **6**.

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