Reverse divisible number
{{Short description|Integers that evenly divide their digit reversal}}
In number theory, reversing the digits of a number {{mvar|n}} sometimes produces another number {{mvar|m}} that is divisible by {{mvar|n}}.
This happens trivially when {{mvar|n}} is a palindromic number; the nontrivial reverse divisors are
:1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... {{OEIS|A008919}}.
For instance, 1089 × 9 = 9801, the reversal of 1089, and 2178 × 4 = 8712, the reversal of 2178.{{citation
| last1 = Webster | first1 = R.
| last2 = Williams | first2 = G.
| issue = 3
| journal = Mathematical Spectrum
| pages = 96–102
| title = On the trail of reverse divisors: 1089 and all that follow
| url = http://users.mct.open.ac.uk/gw3285/publications/reverse-divisors.pdf
| volume = 45
| year = 2013}}.{{citation
| last = Sloane | first = N. J. A. | author-link = Neil Sloane
| arxiv = 1307.0453
| journal = Fibonacci Quarterly
| pages = 99–120
| title = 2178 and all that
| volume = 52
| year = 2014| bibcode = 2013arXiv1307.0453S}}.{{citation
| last1 = Grimm | first1 = C. A.
| last2 = Ballew | first2 = D. W.
| journal = Journal of Recreational Mathematics
| pages = 89–91
| title = Reversible multiples
| volume = 8
| year = 1975–1976}}. As cited by {{harvtxt|Sloane|2014}}.{{citation
| last1 = Klosinski | first1 = L. F.
| last2 = Smolarski | first2 = D. C.
| doi = 10.2307/2688542
| journal = Mathematics Magazine
| pages = 208–210
| title = On the reversing of digits
| volume = 42
| year = 1969| issue = 4
| jstor = 2688542
}}.
The multiples produced by reversing these numbers, such as 9801 or 8712, are sometimes called palintiples.{{citation
| last = Holt | first = Benjamin V.
| journal = Integers
| mr = 3256704
| page = A42
| title = Some general results and open questions on palintiple numbers
| url = http://www.emis.de/journals/INTEGERS/papers/o42/o42.Abstract.html
| volume = 14
| year = 2014}}.
Properties
Every nontrivial reverse divisor must be either 1/4 or 1/9 of its reversal.
The number of {{mvar|d}}-digit nontrivial reverse divisors is where denotes the {{mvar|i}}th Fibonacci number.
For instance, there are two four-digit reverse divisors, matching the formula .{{Cite OEIS|A008919}}
History
The reverse divisor properties of the first two of these numbers, 1089 and 2178, were mentioned by W. W. Rouse Ball in his Mathematical Recreations.{{citation
| last = Ball | first = W. W. Rouse | author-link = W. W. Rouse Ball
| page = 12
| publisher = Macmillan
| title = Mathematical Recreations and Essays
| url = https://archive.org/stream/mathematicalrecr00ball#page/12
| year = 1914}}. In A Mathematician's Apology, G. H. Hardy criticized Rouse Ball for including this problem, writing:
:"These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to a mathematician. The proofs are neither difficult nor interesting—merely tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and proofs, which are not capable of any significant generalization."{{citation|title= A Mathematician's Apology
|author=G. H. Hardy|publisher= Cambridge University Press|year= 2012|isbn= 9781107604636|page=105|url=https://books.google.com/books?id=EkY2im6xkVkC&pg=PA105}}.
References
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