Find the unit digit of the expression
`31^(2)+32^(2)+33^(2)+34^(2)+35^(2)+36^(2)+37^(2)+38^(2)+39^(2)`.

Options

A. 1
B. 4
C. 5
D. 9

Answer

To find the unit digit of the expression \(31^2 + 32^2 + 33^2 + 34^2 + 35^2 + 36^2 + 37^2 + 38^2 + 39^2\), we can follow these steps:

### Step 1: Identify the unit digits of each number
The unit digits of the numbers from 31 to 39 are:
- \(31 \rightarrow 1\)
- \(32 \rightarrow 2\)
- \(33 \rightarrow 3\)
- \(34 \rightarrow 4\)
- \(35 \rightarrow 5\)
- \(36 \rightarrow 6\)
- \(37 \rightarrow 7\)
- \(38 \rightarrow 8\)
- \(39 \rightarrow 9\)

### Step 2: Calculate the squares of the unit digits
Next, we calculate the squares of these unit digits:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\) (unit digit is \(6\))
- \(5^2 = 25\) (unit digit is \(5\))
- \(6^2 = 36\) (unit digit is \(6\))
- \(7^2 = 49\) (unit digit is \(9\))
- \(8^2 = 64\) (unit digit is \(4\))
- \(9^2 = 81\) (unit digit is \(1\))

### Step 3: List the unit digits of the squares
The unit digits of the squares are:
- From \(31^2\): \(1\)
- From \(32^2\): \(4\)
- From \(33^2\): \(9\)
- From \(34^2\): \(6\)
- From \(35^2\): \(5\)
- From \(36^2\): \(6\)
- From \(37^2\): \(9\)
- From \(38^2\): \(4\)
- From \(39^2\): \(1\)

### Step 4: Sum the unit digits
Now, we sum these unit digits:
\[ 1 + 4 + 9 + 6 + 5 + 6 + 9 + 4 + 1 \]

Calculating this step-by-step:
- \(1 + 4 = 5\)
- \(5 + 9 = 14\)
- \(14 + 6 = 20\)
- \(20 + 5 = 25\)
- \(25 + 6 = 31\)
- \(31 + 9 = 40\)
- \(40 + 4 = 44\)
- \(44 + 1 = 45\)

### Step 5: Find the unit digit of the total sum
The unit digit of \(45\) is \(5\).

### Final Answer
Thus, the unit digit of the expression \(31^2 + 32^2 + 33^2 + 34^2 + 35^2 + 36^2 + 37^2 + 38^2 + 39^2\) is **5**.

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