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

Step 1: Digits \( 1 \) to \( 9 \) use \( 9 \) digits; the remaining \( 41-9=32 \) digits come from two-digit numbers starting at \( 10 \), covering \( 16 \) numbers: \( 10 \) through \( 25 \).

Step 2: So the \( 41 \)-digit string ends in \( \ldots2425 \), and only the last three digits, \( 425 \), determine the remainder mod \( 8 \) (since \( 1000 \) is a multiple of \( 8 \)).

Step 3: Dividing, \( 425=8\times53+1 \).
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