Find Unit digit of `2^(20!).`

Answer

To find the unit digit of \(2^{20!}\), we can follow these steps:

### Step 1: Understand the cyclicity of the unit digits of powers of 2.
The unit digits of the powers of 2 follow a specific pattern:
- \(2^1 = 2\) (unit digit is 2)
- \(2^2 = 4\) (unit digit is 4)
- \(2^3 = 8\) (unit digit is 8)
- \(2^4 = 16\) (unit digit is 6)
- \(2^5 = 32\) (unit digit is 2)
- \(2^6 = 64\) (unit digit is 4)

From this, we can see that the unit digits repeat every 4 powers: 2, 4, 8, 6.

### Step 2: Determine the power modulo 4.
To find the unit digit of \(2^{20!}\), we need to find \(20! \mod 4\).

### Step 3: Calculate \(20!\).
Since \(20!\) is the product of all integers from 1 to 20, it includes multiple factors of 2. In fact, \(20!\) contains many multiples of 4, which means \(20! \mod 4 = 0\).

### Step 4: Use the cyclicity to find the unit digit.
Since \(20! \mod 4 = 0\), we look at the unit digit corresponding to \(2^0\) in the cycle of unit digits:
- \(2^1\) has a unit digit of 2
- \(2^2\) has a unit digit of 4
- \(2^3\) has a unit digit of 8
- \(2^4\) has a unit digit of 6

When the exponent is a multiple of 4, the unit digit is 6.

### Conclusion:
Thus, the unit digit of \(2^{20!}\) is **6**.

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