Find the remainder when the $41$-digit number $1234\ldots$ is divided by $8$.
Find the remainder when the $41$-digit number $1234\ldots$ is divided by $8$.
Show Hint
- $1$
- $2$
- $3$
$4$
The Correct Option is A
Solution and Explanation
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 $1$.
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