Let E(n) denote the sum of the even digits of n. For example, `E (1243)=2 +4 =6.` What is the value of `E(1) + E(2) + E(3)+…….+ E(100)?`

Answer

To find the value of \( E(1) + E(2) + E(3) + \ldots + E(100) \), we will break down the problem into two parts: the contribution from the unit digits and the contribution from the tens digits.

### Step 1: Calculate the contribution from unit digits

The even digits are 0, 2, 4, 6, and 8. We will calculate the sum of these even digits for all numbers from 1 to 100.

1. **Sum of even digits from 1 to 9:**
- The even digits are: 2, 4, 6, 8.
- Their sum is: \( 2 + 4 + 6 + 8 = 20 \).

2. **Count how many times this occurs:**
- The unit digits repeat every 10 numbers (i.e., 1-9, 11-19, ..., 91-99).
- From 1 to 99, there are 10 complete sets of 10 (0-9, 10-19, ..., 90-99).
- Therefore, the contribution from the unit digits is:
\[ 20 \times 10 = 200. \]

### Step 2: Calculate the contribution from tens digits

Next, we will calculate the contribution from the tens digits.

1. **Identify the even digits in the tens place:**
- The even digits are: 0, 2, 4, 6, and 8.
- The ranges for these digits are:
- 0: from 1 to 9 (10 numbers)
- 2: from 20 to 29 (10 numbers)
- 4: from 40 to 49 (10 numbers)
- 6: from 60 to 69 (10 numbers)
- 8: from 80 to 89 (10 numbers)

2. **Calculate the contribution from each even digit in the tens place:**
- For 0: \( 0 \times 10 = 0 \)
- For 2: \( 2 \times 10 = 20 \)
- For 4: \( 4 \times 10 = 40 \)
- For 6: \( 6 \times 10 = 60 \)
- For 8: \( 8 \times 10 = 80 \)

3. **Total contribution from the tens digits:**
- Sum these contributions:
\[ 0 + 20 + 40 + 60 + 80 = 200. \]

### Step 3: Combine both contributions

Now, we will add the contributions from the unit digits and the tens digits:
\[ E(1) + E(2) + E(3) + \ldots + E(100) = 200 + 200 = 400. \]

### Final Answer
Thus, the value of \( E(1) + E(2) + E(3) + \ldots + E(100) \) is \( \boxed{400} \).