Understanding Divisibility by 9 for Repeated Digits

The question asks for the least number of digits, \(n\), such that a number consisting of \(n\) ones (111...1) is divisible by 9.

Divisibility Rule for 9

A fundamental rule in number theory states that a number is divisible by 9 if and only if the sum of its digits is divisible by 9. This is a crucial concept for solving this problem.

Applying the Rule to the Given Number

The number in question is formed by repeating the digit 1 exactly \(n\) times. Let's analyze the sum of its digits:

• If \(n=1\), the number is 1. Sum of digits = 1.
• If \(n=2\), the number is 11. Sum of digits = 1 + 1 = 2.
• If \(n=3\), the number is 111. Sum of digits = 1 + 1 + 1 = 3.
• In general, if the number has \(n\) digits, all of which are 1, the sum of the digits is \(1 + 1 + \dots + 1\) (n times).

So, the sum of the digits of the number consisting of \(n\) ones is \(n \times 1 = n\).

Condition for Divisibility by 9

According to the divisibility rule for 9, the number consisting of \(n\) ones is divisible by 9 if and only if the sum of its digits, which is \(n\), is divisible by 9.

We are looking for the least value of \(n\) such that \(n\) is divisible by 9.

Finding the Least Value of \(n\)

The positive integers that are divisible by 9 are 9, 18, 27, 36, and so on (multiples of 9). The smallest positive integer among these is 9.

Therefore, the least value of \(n\) for which the number 111...1 (n digits) is divisible by 9 is 9.

Let's check this:

• If \(n=9\), the number is 111,111,111. The sum of the digits is \(1 \times 9 = 9\). Since 9 is divisible by 9, the number 111,111,111 is divisible by 9.
• If we take a smaller value of \(n\) like 3 (from the options), the number is 111. The sum of digits is 3. 111 is not divisible by 9.

Analyzing the Options

Let's examine the given options:

• Option 1: 9 - If \(n=9\), sum of digits is 9. 9 is divisible by 9. This is a valid value for \(n\).
• Option 2: 18 - If \(n=18\), sum of digits is 18. 18 is divisible by 9. This is a valid value for \(n\), but not the least.
• Option 3: 3 - If \(n=3\), sum of digits is 3. 3 is not divisible by 9. This is not a valid value for \(n\).
• Option 4: 12 - If \(n=12\), sum of digits is 12. 12 is not divisible by 9. This is not a valid value for \(n\).

Comparing the valid options (9 and 18), the least value is 9.

Conclusion

The least value of \(n\) for which the number 111...1 (n digits) is divisible by 9 is 9.

Revision Table: Divisibility Rules for 9 and 3

Understanding divisibility rules is key to solving problems like this quickly. The rule for 9 is closely related to the rule for 3.

Additional Information: Properties of Repunits

Numbers consisting only of the digit 1 are sometimes called repunits. A repunit with \(n\) digits is often denoted as \(R_n\). We found that \(R_n\) is divisible by 9 if \(n\) is a multiple of 9.

Similarly, we can explore other divisibility properties for repunits:

• \(R_n\) is divisible by 3 if \(n\) is a multiple of 3 (since if \(n\) is a multiple of 3, the sum of digits \(n\) is a multiple of 3).
• \(R_n\) is divisible by 11 if \(n\) is even.

These properties stem from the structure of the numbers and divisibility rules.