Finding the Least 4-Digit Number Divisible by 9

The question asks us to find the smallest possible number that has exactly four digits and can be divided by $9$ without leaving any remainder.

Understanding the Requirements

• Least 4-digit number: The smallest integer composed of four digits. This number is $1000$.
• Exactly divisible by 9: The number must be a multiple of $9$. We can use the divisibility rule for $9$ to check this.

Step-by-Step Solution

We need to start checking from the smallest 4-digit number, which is $1000$, and find the first one that is divisible by $9$.

Method 1: Using the Divisibility Rule of 9

A number is divisible by $9$ if the sum of its digits is divisible by $9$. Let's check the numbers starting from $1000$:

• Number: $1000$
• Sum of digits: $1 + 0 + 0 + 0 = 1$. Since $1$ is not divisible by $9$, $1000$ is not divisible by $9$.

We continue checking subsequent numbers:

• Number: $1001$, Sum = $1+0+0+1 = 2$ (Not divisible by $9$)
• Number: $1002$, Sum = $1+0+0+2 = 3$ (Not divisible by $9$)
• Number: $1003$, Sum = $1+0+0+3 = 4$ (Not divisible by $9$)
• Number: $1004$, Sum = $1+0+0+4 = 5$ (Not divisible by $9$)
• Number: $1005$, Sum = $1+0+0+5 = 6$ (Not divisible by $9$)
• Number: $1006$, Sum = $1+0+0+6 = 7$ (Not divisible by $9$)
• Number: $1007$, Sum = $1+0+0+7 = 8$ (Not divisible by $9$)
• Number: $1008$
• Sum of digits: $1 + 0 + 0 + 8 = 9$. Since $9$ is divisible by $9$, the number $1008$ is divisible by $9$.

Since $1008$ is the first 4-digit number we found that satisfies the condition, it is the least 4-digit number exactly divisible by $9$.

Method 2: Using Division and Remainder

We can find the smallest 4-digit number by dividing $1000$ by $9$ and seeing what the remainder is.

Calculate the division:

Performing the division:

The quotient is $111$ and the remainder is $1$. This means $1000$ is $1$ more than a multiple of $9$ (which is $999$).

To find the next multiple of $9$ (which will be the smallest 4-digit multiple), we need to add the difference between the divisor ($9$) and the remainder ($1$) to the original number ($1000$).

Difference needed = $9 - \text{remainder} = 9 - 1 = 8$.

The least 4-digit number divisible by $9$ is $1000 + 8 = 1008$.

Conclusion

Both methods confirm that $1008$ is the smallest 4-digit number that is exactly divisible by $9$.