A $99$-digit number is formed by writing the first $59$ natural numbers one after the other as $1234567891011121314\dots5859$. Find the remainder obtained when the above number is divided by $16$.

### Concept & Logic To check for divisibility by powers of $2$, you only need to examine the last digits of the number. * Divisibility by $2^1$ ($2$): Check the last $1$ digit. * Divisibility by $2^2$ ($4$): Check the last $2$ digits. * Divisibility by $2^3$ ($8$): Check the last $3$ digits. * Divisibility by $2^4$ ($16$): Check the last $4$ digits. ### Step-by-Step Solution **Given:** The number is $12345678910\dots575859$. The divisor is $16$. **Calculation / Deduction:** * Since $16 = 2^4$, the remainder of the entire $99$-digit number divided by $16$ will be exactly the same as the remainder of just its last $4$ digits divided by $16$. * The sequence ends with $57, 58, 59$. The last four digits form the number $5859$. * Now, simply divide $5859$ by $16$. $$ 5859 = 16 \times 366 + 3 $$ * Let's break down the division mentally: * $16 \times 300 = 4800$ (Remainder: $1059$) * $16 \times 60 = 960$ (Remainder: $99$) * $16 \times 6 = 96$ (Remainder: $3$) * The final remainder is $3$. ### Exam Strategy & Shortcut Do not let the "$99$-digit number" phrase intimidate you; it is just filler. Identify the divisor ($16$), recall the rule of $16$ (last $4$ digits), extract those digits ($5859$), and perform standard division. ### Common Pitfall A common mistake is attempting to use modular arithmetic on blocks of numbers or trying to find a pattern in the remainders of $1, 2, 3\dots$, which is a massive waste of time and will not yield the correct answer. Stick to the basic divisibility rules. ### Final Answer Therefore, the correct answer is 3.