Why is the Difference Between a Number and its Rearrangement Always Divisible by 9?

Why is the Difference Between a Number and its Rearrangement Always Divisible by 9?

Have you ever noticed that if you rearrange the digits of a number and subtract the smaller number from the larger one, the result is always divisible by 9? This intriguing mathematical phenomenon has fascinated mathematicians for centuries, and the explanation lies in the fascinating world of number theory.

Understanding the Concept

Let's break down this concept with an example. Consider the number 1234. Now, let's rearrange the digits to form 4321. Subtracting the smaller number from the larger one, we get:

4321 - 1234 = 3087

Notice that 3087 is perfectly divisible by 9. This is not a coincidence; it's a mathematical rule that holds true for any number and its rearrangement.

The Mathematical Proof

The reason behind this phenomenon lies in the concept of place value and divisibility rules. Every digit in a number represents a power of 10. For instance, in the number 1234:

• 1 is in the thousands place (10^3)
• 2 is in the hundreds place (10^2)
• 3 is in the tens place (10^1)
• 4 is in the units place (10^0)

When you rearrange the digits, you're essentially changing the place value of each digit. However, the sum of the digits remains the same. This is because the sum of the digits of a number is equal to the remainder when the number is divided by 9. For example, the sum of the digits of 1234 is 10, and the remainder when 1234 is divided by 9 is also 10.

When you subtract the smaller rearranged number from the larger one, you're essentially subtracting the same sum of digits from itself. This difference will always be a multiple of 9, making it divisible by 9.

Example

Let's consider another example: 5678.

Rearranging the digits, we get 8765.

Subtracting the smaller number from the larger one:

8765 - 5678 = 3087

As you can see, 3087 is divisible by 9.

Conclusion

The divisibility by 9 rule is a fascinating mathematical concept that demonstrates the power of number theory. It highlights the relationship between place value, divisibility, and the sum of digits. Understanding this principle can make it easier to identify numbers divisible by 9 and can be a valuable tool in various mathematical applications.