Palintiples: Numbers Divisible by Their Reversed Digits

Are there numbers that, when their digits are reversed, can be evenly divided by the original number? This intriguing mathematical concept has lately captured the interest of many. These numbers, known as palintiples, are indeed interesting and can be generated through specific rules and transformations.

What are Palintiples?

A palintiple is a positive integer whose reverse is divisible by the number itself. For instance, the number 8712 is a palintiple because when its digits are reversed, the number 2178 can be evenly divided by 8712. Similarly, 9801 is also a palintiple as 1089 divides into 9801.

Examples of Palintiples

Here are some of the smallest palintiples:

8712 4 × 2178 9801 9 × 1089

These numbers, like any palintiple, can be created using a few simple transformations. For example, inserting a 9 between the digits of a palintiple produces another palintiple. Moreover, concatenating two identical palintiples with zeros inserted in between also creates a palintiple. Thus, numbers like 98999901 and 879120087912 are palintiples. This method of generating palintiples was first noted by Hoey.

Theorems and Transformations

Investigating palintiples goes beyond simple numerical transformations. Let us explore more in-depth:

The product of a palintiple and a specific integer (like 9 or 4 in certain cases) will always result in its reverse. There are infinitely many palintiples, and one can generate new ones using the given number and simple rules, like inserting a 9 or concatenating with 0s in between.

Mathematical Puzzles and Solutions

Many mathematical puzzles revolve around palintiples. For example, try to find a four-digit number whose reverse is equal to the original number multiplied by 4. The solution to this specific puzzle is 2178 (2178 × 4 8712).

Proof and Generalization

In general, if a number ( N ) with the digits ABCD is a palintiple, then:

ABCD x 4 DCBA ABCD x 9 DCBA

By analyzing the possible digit combinations, one can construct the prototypes 1089 and 2178. Further generalization shows that a palintiple can be represented as follows:

1099...989 × 9 9899...901 (where there are n number of 9's) 2199...978 × 4 8799...912 (where there are n number of 9's)

By stringing these components together in a palindromic manner, a wide range of palintiples can be generated.

Conclusion

In conclusion, palintiples not only provide a fascinating challenge for puzzle enthusiasts but also reveal interesting patterns in number theory. While these results are not as glamorous as some might claim, they still offer a unique and engaging problem in mathematics.

Keywords: palintiple, number reverse, mathematical puzzle