The unit digit of `2^(3^4) xx 3^(4^5) xx 4^(5^6) xx 5^(6^7) xx 6^(7^8) xx 7^(8^9)` is :

Options

A. 0
B. 5
C. can't be determined
D. none of these

Answer

To find the unit digit of the expression \(2^{(3^4)} \times 3^{(4^5)} \times 4^{(5^6)} \times 5^{(6^7)} \times 6^{(7^8)} \times 7^{(8^9)}\), we will analyze the unit digits of each component separately.

### Step 1: Find the unit digit of \(2^{(3^4)}\)
1. Calculate \(3^4\):
\[ 3^4 = 81 \]
2. The unit digits of powers of 2 cycle every 4: \(2, 4, 8, 6\).
- \(2^1 \equiv 2\)
- \(2^2 \equiv 4\)
- \(2^3 \equiv 8\)
- \(2^4 \equiv 6\)
- Then it repeats.
3. To find \(2^{81}\), we find \(81 \mod 4\):
\[ 81 \mod 4 = 1 \]
Thus, the unit digit of \(2^{81}\) is the same as \(2^1\), which is **2**.

### Step 2: Find the unit digit of \(3^{(4^5)}\)
1. Calculate \(4^5\):
\[ 4^5 = 1024 \]
2. The unit digits of powers of 3 cycle every 4: \(3, 9, 7, 1\).
3. To find \(3^{1024}\), we find \(1024 \mod 4\):
\[ 1024 \mod 4 = 0 \]
Thus, the unit digit of \(3^{1024}\) is the same as \(3^4\), which is **1**.

### Step 3: Find the unit digit of \(4^{(5^6)}\)
1. Calculate \(5^6\):
\[ 5^6 = 15625 \]
2. The unit digits of powers of 4 cycle every 2: \(4, 6\).
3. To find \(4^{15625}\), we find \(15625 \mod 2\):
\[ 15625 \mod 2 = 1 \]
Thus, the unit digit of \(4^{15625}\) is the same as \(4^1\), which is **4**.

### Step 4: Find the unit digit of \(5^{(6^7)}\)
1. Calculate \(6^7\):
\[ 6^7 = 279936 \]
2. The unit digit of any power of 5 is always **5**.

### Step 5: Find the unit digit of \(6^{(7^8)}\)
1. Calculate \(7^8\):
\[ 7^8 = 5764801 \]
2. The unit digit of any power of 6 is always **6**.

### Step 6: Find the unit digit of \(7^{(8^9)}\)
1. Calculate \(8^9\):
\[ 8^9 = 134217728 \]
2. The unit digits of powers of 7 cycle every 4: \(7, 9, 3, 1\).
3. To find \(7^{134217728}\), we find \(134217728 \mod 4\):
\[ 134217728 \mod 4 = 0 \]
Thus, the unit digit of \(7^{134217728}\) is the same as \(7^4\), which is **1**.

### Step 7: Combine the unit digits
Now we multiply the unit digits we found:
- From \(2^{(3^4)}\): **2**
- From \(3^{(4^5)}\): **1**
- From \(4^{(5^6)}\): **4**
- From \(5^{(6^7)}\): **5**
- From \(6^{(7^8)}\): **6**
- From \(7^{(8^9)}\): **1**

Calculating the unit digit of the product:
\[ 2 \times 1 \times 4 \times 5 \times 6 \times 1 \]
Calculating step-by-step:
1. \(2 \times 1 = 2\)
2. \(2 \times 4 = 8\)
3. \(8 \times 5 = 40\) (unit digit is **0**)
4. \(0 \times 6 = 0\)
5. \(0 \times 1 = 0\)

Thus, the final unit digit of the entire expression is **0**.