Question

Find the remainder when the $41$-digit number $1234\ldots$ is divided by $8$. 

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Hint:

Divisibility by $8$ depends only on the last $3$ digits of the number.

Answer 1

The Correct Option is A

Interpreting $1234\ldots$ as the string $1234567891011\ldots$ continued until $41$ digits. Only the \emph{last three} digits matter mod $8$. Digits $1$–$9$ use $9$ places; remaining $32$ places are from two-digit numbers. That is $16$ numbers: $10$ to $25$. The final three digits are the last digit of $24$ and both digits of $25$, i.e. $425$. \[ 425 \div 8=53 \text{ remainder } 1. \] So the remainder is $1$. 

Answer 2

Since \( 1000 \) is divisible by \( 8 \), every digit in a number's thousands place or higher contributes a multiple of \( 8 \) when the number is expanded by place value; hence the remainder mod \( 8 \) depends only on the last three digits. First pin down exactly which digits form the end of the \( 41 \)-digit string \( 123456789101112\ldots \).

Count digits used: the numbers \( 1 \) through \( 9 \) use \( 9\times1=9 \) digits. Remaining digits needed: \( 41-9=32 \), and since each number from \( 10 \) onward uses \( 2 \) digits, this covers \( 32/2=16 \) two-digit numbers, i.e. \( 10,11,\ldots,25 \). So the string ends exactly at \( \ldots24\,25 \), and its last three digits are \( 4,2,5 \), i.e. \( 425 \).

1. Option A (\( 1 \)): Dividing, \( 425=8\times53+1 \), giving remainder \( 1 \). This matches.
2. Option B (\( 2 \)): Checking \( 425=8\times53+1 \), the remainder is \( 1 \), not \( 2 \); rejected.
3. Option C (\( 3 \)): Again the remainder is \( 1 \), not \( 3 \); rejected.
4. Option D (\( 4 \)): The remainder is \( 1 \), not \( 4 \); rejected.

Deriving from first principles why only the last three digits matter, then correctly locating them at \( 425 \), gives a remainder of \( 1 \) upon dividing by \( 8 \).

Hence, the correct answer is 1.