Finding the Smallest 4-Digit Number Divisible by 19: A Comprehensive Guide

Let's explore the process of finding the smallest 4-digit number that is exactly divisible by 19. This exploration will not only answer this specific question but also provide a method that can be applied to similar problems.

Understanding the Concept

Divisibility by 19 means that when a number is divided by 19, the remainder should be zero. In other words, the number must be a multiple of 19. When dealing with 4-digit numbers, the smallest such number can be identified by following a step-by-step approach.

Step-by-Step Guide

Let's break down the process to find the smallest 4-digit number divisible by 19:

Start with the smallest 4-digit number: The smallest 4-digit number is 1000. Divide 1000 by 19: When 1000 is divided by 19, the quotient is 52 with a remainder of 12. This can be represented as 1000 19 * 52 12. Identify the remainder: The remainder is 12, which means 1000 is 12 less than the next multiple of 19. Calculate the next multiple of 19: To find the next multiple of 19, we need to add the difference between 19 and the remainder to 1000. So, 1000 - 12 19 1007.

Verification

The calculation can be verified as follows:

Check the divisibility: 1007 ÷ 19 53, which is a whole number, indicating that 1007 is exactly divisible by 19. Confirm the remainder: When 1000 is divided by 19, the remainder is 12. Adding 19 to 1000 (1000 19) would make the remainder 0, confirming that 1007 is the next multiple of 19.

Alternatively, you can use the following approaches to verify:

Subtracting the remainder from 1000: 1000 - 12 988 (a 3-digit number), and the next multiple of 19 is 988 19 1007. Estimating from a rough calculation: If you estimate 1000 as close to 95 (which is 19 * 5), you can adjust by adding 7 to 95 to get 102 (which is closer to 1007). The next multiple of 19 that starts with 5 is 57, and multiplying by 19 gives 1007. Dividing 1007 by 19: 1007 ÷ 19 53, confirming that 1007 is the smallest 4-digit number divisible by 19.

Conclusion

The smallest 4-digit number divisible by 19 is indeed 1007. This problem-solving method can be applied to any similar question involving finding the smallest n-digit number divisible by a specific divisor. By understanding the concept of remainders and multiples, you can systematically determine such numbers.

Related Resources

For further exploration into divisibility rules and number theory, you might find the following resources helpful:

Wikipedia articles on divisibility and remainders. Online calculators and tools for checking divisibility. Math textbooks or online courses on number theory.

By mastering these concepts, you can improve your mathematical skills and perform well in various areas, from competitive exams to real-world problem-solving scenarios.