Solving for x in a 5-Digit Number Divisible by 11 with Specific Digits

Have you ever faced a math problem where you need to find a specific 5-digit number that is divisible by 11, and the digits at certain places are predefined? This article will guide you through solving such a problem step-by-step, providing a clear and detailed explanation.

Understanding the Problem

The problem we are dealing with is to find a 5-digit number, denoted as x, where:

The number contains the digit 1 in the thousands, hundreds, and ten thousands places. The number is divisible by 11.

The Solution Approach

Let's denote the five-digit number x as abcde, where a, b, c, d, and e are the digits from right to left. According to the given conditions:

a c d e 1

The number thus looks like this: ab111. Now, to be divisible by 11, the number must satisfy the divisibility rule for 11, which states that the sum of the digits at the odd places must equal the sum of the digits at the even places.

Applying the Divisibility Rule for 11

The digits in ab111 at odd places are a, 1, 1 and at even places are b, 1. Therefore, we need to find b such that:

a 1 1 b 1

Substituting the value of a 1, we get:

1 1 1 b 1
3 b 1
b 2

So, the number x is 11121.

Verification and Direct Method

To confirm, we can check if 11121 is divisible by 11 by using the same divisibility rule. We have:

1 1 1 3
1 2 3

Since both sums are equal, 11121 is indeed divisible by 11.

The second approach is to list all possible numbers formed by the given conditions and check their divisibility by 11. The possible 5-digit numbers are:

11101 11111 11121 11131 11141 11151 11161 11171 11181 11191

Applying the divisibility rule to each, we find that 11121 is the only number that satisfies the condition.

Therefore, the number x is 11121.

Conclusion

In summary, to solve for x in a 5-digit number that is divisible by 11 and has specific digits, we can either apply the divisibility rule for 11 directly or list and test all possible numbers. The direct method we used here provides a clear and efficient solution.

Have you tried any other direct methods to solve this problem? Share your approach in the comments below! Thanks for reading!