Introduction

In mathematics, we often encounter problems where we need to find the largest number that meets specific criteria. In this article, we will explore a technique to find the largest four-digit number that is divisible by 7, using both programming and mathematical methods. We will discuss the process using the J programming language, as well as providing a mathematical explanation for the trick behind a specific multiplication pattern.

Using the J Programming Language

The J programming language provides a straightforward way to solve this problem through a brute force method. Here's how you can do it:

Use the range notation to specify the four-digit numbers from 9990 to 9999. Check for divisibility by 7 among these numbers. Identify the largest number that satisfies the condition.

~079990 to 9999
9996

The result is 9996. This number is the largest four-digit number divisible by 7.

Mathematical Trick for Divisibility by 7

There is a neat trick in mathematics where if you multiply any three-digit number by 7, 11, and 13, the result is a six-digit number. The first three digits and the last three digits of this six-digit number are the same as the original three-digit number. Let's break down why this works.

First, consider the multiplication of these numbers:

7*11*131001. Multiplying 1001 by any three-digit number will result in a six-digit number where the first three digits and the last three digits are the same as the original number. For example:

117 * 7 * 11 * 13  117117
523 * 7 * 11 * 13  523523
945 * 7 * 11 * 13  945945

Now, let's apply this insight to find the largest four-digit number divisible by 7:

Use the number 1001 and apply it to a four-digit number. Subtract 14 from 10010 to get the largest four-digit number divisible by 7. The result is 9996, which we can verify by dividing it by 7:

9996 / 7  1428

Since 9996 is divisible by 7 and is a four-digit number, we can conclude that 9996 is the largest four-digit number that meets the criteria.

Multiplication Pattern Explained

Let's explore the multiplication pattern in more detail:

Take the number 1001. Multiply it by 10 to get 10010. Notice that 10010 is 14 more than the largest four-digit number, 9996. To obtain the largest four-digit number divisible by 7, subtract 14 from 10010.

10010 - 14  9996

This process ensures that 9996 is the largest four-digit number that is divisible by 7.

Using Division to Find the Answer

An alternative approach involves simple division:

Divide 9999 by 7 to get 1428.4. Multiply 1428 by 7 to get 9996.

The result is 9996, which is confirmed to be the largest four-digit number divisible by 7 through both the multiplication trick and simple division.

Conclusion

Through various methods, including mathematical tricks and programming techniques, we have determined that the largest four-digit number divisible by 7 is 9996. These techniques not only provide the answer but also illustrate the elegance and power of mathematical and computational approaches in problem-solving.