The reversal of a positive integer 
is 
. The reversal of a positive
integer 
is implemented in the Wolfram Language
as IntegerReverse[n].
A positive integer that is the same as its own reversal is known as a palindromic number.
More generally, if 
is a positive integer having 
base-
digits, its reversal is 
. Since 
, it follows that 
, so 
is a multiple of 
. In particular, for the two-digit decimal integer 
, 
. Similarly, 
gives 
.
Ball and Coxeter (1987) consider numbers whose reversals are integral multiples of themselves. Palindromic numbers and numbers ending with a zero are trivial examples.
The first few nontrivial examples of numbers whose reversals are multiples of themselves are 8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901, 87128712,
87999912, 98019801, 98999901, ... (OEIS A031877).
The pattern continues for large numbers, with numbers of
the form 
equal to 4 times their reversals and numbers of the form 
equal to 9 times their reversals.
In addition, runs of numbers of either of these forms can be concatenated
to yield numbers of the form 
, equal to 4 times their reversals,
and 
, equal to
9 times their reversals.
The reversals corresponding to the above are 1089, 2178, 10989, 21978, 109989, 219978, ... (OEIS A008919).
The product of a two-digit number and its reversal is never a square number except when the digits are the same (Ogilvy 1988).
Numbers whose product is the reversal of the products of their reversals include (221, 312) and (122, 213), since
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(1)
|
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(2)
|
(Ball and Coxeter 1987, p. 14).
Non-palindromic numbers 
such that 
is not divisible by 10 and 
is square,
where 
is the reversal of 
, are given by 144, 169, 288, 441, 528, ... (OEIS A062917).
The only known perfect powers with exponent greater than 2 resulting from reversal multiplication are
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(3)
|
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(4)
|
