When number 1 2 3 4 5 ................. (76 digits) is divided by 16, the remainder will be?

Options

A. 7
B. 0
C. 6
D. 2

Answer

To solve the problem of finding the remainder when the number formed by the digits 1, 2, 3, ..., up to 76 digits is divided by 16, we can follow these steps:

### Step 1: Identify the number of digits
The sequence starts with 1 and continues sequentially. We need to determine how many digits are in the sequence up to 76.

- The digits from 1 to 9 contribute 9 digits.
- The digits from 10 to 99 contribute 2 digits each.

To find how many complete numbers we can have with 76 digits:
- From 1 to 9: 9 digits (1 to 9)
- From 10 onwards: Each number contributes 2 digits.

Let’s calculate how many more digits we need after 9:
- Total digits needed = 76
- Digits already counted = 9
- Remaining digits = 76 - 9 = 67

Now, we can find how many complete two-digit numbers can be formed with 67 digits:
- Each two-digit number contributes 2 digits, so the number of two-digit numbers we can have is 67 / 2 = 33.5.

Since we can only have whole numbers, we take 33 complete two-digit numbers.

### Step 2: Calculate the total numbers used
- From 1 to 9: 9 digits (1 to 9)
- From 10 to 42: 33 numbers (10 to 42 contributes 66 digits)

Thus, the digits used so far:
- Total digits = 9 (from 1 to 9) + 66 (from 10 to 42) = 75 digits.

### Step 3: Identify the last digit
Since we need 76 digits, the next digit after 42 is 43, which will be the 76th digit.

### Step 4: Determine the last four digits
The last four digits of the number formed by the sequence are:
- 41, 42, 43

So, the last four digits are **41, 42, 43, 44**.

### Step 5: Calculate the remainder when divided by 16
Now we need to find the remainder of the number 4243 when divided by 16.

1. Divide 4243 by 16:
- 16 × 264 = 4224
- 4243 - 4224 = 19

So, the remainder when 4243 is divided by 16 is **19**.

### Final Answer
The remainder when the number formed by the digits 1 to 76 is divided by 16 is **19**.

---