In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, reversing the digits of a number sometimes produces another number that is divisible by .
This happens trivially when is a
palindromic number
A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ...
; the nontrivial reverse divisors are
:1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... .
For instance, 1089 × 9 = 9801, the reversal of 1089, and 2178 × 4 = 8712, the reversal of 2178.
[.][.]
The multiples produced by reversing these numbers, such as 9801 or 8712, are sometimes called palintiples.
Properties
Every nontrivial reverse divisor must be either 1/4 or 1/9 of its reversal.
The number of -digit nontrivial reverse divisors is
where
denotes the th
Fibonacci number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
.
For instance, there are two four-digit reverse divisors, matching the formula
.
History
The reverse divisor properties of the first two of these numbers, 1089 and 2178, were mentioned by
W. W. Rouse Ball
Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...
in his ''Mathematical Recreations''. In ''
A Mathematician's Apology
''A Mathematician's Apology'' is a 1940 essay by British mathematician G. H. Hardy which defends the pursuit of mathematics for its own sake. Central to Hardy's "apology" – in the sense of a formal justification or defence (as in Plato's '' ...
'',
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
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."
[.]
References
{{reflist
Base-dependent integer sequences