The numbers from $1$ to $29$ are written side by side as follows: $1234567891011121314..........2829$ If this number is divided by $9$, then what is the remainder?

### Concept & Rule This problem utilizes the **Divisibility Rule of 9**. A number is perfectly divisible by $9$ if the sum of its digits is divisible by $9$. Furthermore, the remainder obtained when dividing a number by $9$ is strictly equal to the remainder obtained when dividing the *sum of its digits* by $9$. ### Step-by-Step Solution * **Given:** * A large number formed by writing $1$ through $29$ sequentially. * We need to find the remainder when this massive number is divided by $9$. * **Calculation:** * We must calculate the total sum of all the individual digits from $1$ to $29$. * **Group 1 (Single digits 1 to 9):** Sum = $1 + 2 + 3 + ... + 9 = 45$ * **Group 2 (Two-digit numbers 10 to 19):** Each number has a ten's digit of $1$. There are ten such numbers. Sum of ten's digits = $10 \times 1 = 10$ Sum of unit's digits ($0$ to $9$) = $45$ Total sum for this group = $10 + 45 = 55$ * **Group 3 (Two-digit numbers 20 to 29):** Each number has a ten's digit of $2$. There are ten such numbers. Sum of ten's digits = $10 \times 2 = 20$ Sum of unit's digits ($0$ to $9$) = $45$ Total sum for this group = $20 + 45 = 65$ * **Total Sum of all digits:** $$\text{Total Sum} = 45 + 55 + 65 = 165$$ * Now, divide this total sum by $9$ to find the remainder: $165 \div 9$ gives a quotient of $18$ ($18 \times 9 = 162$). $$165 - 162 = 3$$ * The remainder is $3$. ### Exam Strategy & Shortcut You can apply the divisibility rule recursively to speed things up. Once you get the sum $165$, you don't even need to divide it by $9$. Just sum the digits of $165$ again: $1 + 6 + 5 = 12$. Sum the digits of $12$ again: $1 + 2 = 3$. The final single digit is your remainder. ### Common Pitfall The most common mistake is attempting to sum the numbers themselves (e.g., using the arithmetic progression sum formula $N(N+1)/2$ to find the sum of $1+2+...+29$) rather than summing the *individual digits* that make up the number string. The rule of $9$ applies strictly to the individual digits. ### Final Answer **Therefore, the correct answer is 3.**