What is the difference between a two-digit number and the number obtained by reversing its digits is always divisible by?

The difference between a two-digit number and the number obtained by reversing its digits is always divisible by 9.

The Difference Between a Two-Digit Number and Its Reverse is Always Divisible By What Number?

The exact answer is 9. The difference between any two-digit number and the number formed by reversing its digits will always be a multiple of nine, meaning it is always divisible by 9.

Understanding the Mathematical Principle

To demonstrate why this is true, we can use a simple algebraic representation:

Represent a Two-Digit Number:
Let a two-digit number be represented as 10a + b, where:
- a is the digit in the tens place (and a must be an integer from 1 to 9).
- b is the digit in the units place (and b must be an integer from 0 to 9).
- Example: If the number is 53, then a = 5 and b = 3.
Represent the Reversed Number:
When you reverse the digits, the new number obtained is 10b + a.
- Example: For 53, the reversed number is 35, where b = 3 and a = 5.
Calculate the Difference:
Now, let's find the difference between the original number and the reversed number:
Difference = (10a + b) - (10b + a)
Simplify the Expression:
Combine like terms:
Difference = 10a - a + b - 10b
Difference = 9a - 9b
Factor Out 9:
You can factor out 9 from the expression:
Difference = 9(a - b)

This final expression 9(a - b) clearly shows that the difference is always a product of 9 and the difference between the tens digit and the units digit (a - b). Since the difference can always be written as 9 multiplied by an integer, it means the difference is inherently divisible by 9.

Illustrative Examples

Let's examine a few examples to solidify this concept:

Original Number	Tens Digit (a)	Units Digit (b)	Reversed Number	Difference (Original - Reversed)	Difference (9 * (a-b))	Divisible by 9?
42	4	2	24	42 - 24 = 18	9 (4 - 2) = 9 2 = 18	Yes
79	7	9	97	79 - 97 = -18	9 (7 - 9) = 9 -2 = -18	Yes
60	6	0	06 (or 6)	60 - 6 = 54	9 (6 - 0) = 9 6 = 54	Yes
88	8	8	88	88 - 88 = 0	9 (8 - 8) = 9 0 = 0	Yes
15	1	5	51	15 - 51 = -36	9 (1 - 5) = 9 -4 = -36	Yes

Practical Insights and Significance

This property is a fundamental concept in basic number theory and algebra, highlighting the elegance of mathematical rules.

Foundation for Divisibility Rules: It demonstrates a specific divisibility rule derived from the place value system.
Mathematical Puzzles: This principle is often used in mathematical riddles and number puzzles, where knowing this property can quickly lead to a solution.
Understanding Place Value: It reinforces the importance of place value in our base-10 number system and how algebraic manipulation can reveal hidden patterns.

For further exploration of divisibility rules and number properties, resources like Brilliant.org offer comprehensive explanations.