The reversal of a positive integer with decimal digits 
is the integer represented by 
, with any leading zeros omitted. 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, where 
is an integer, 
, and 
, 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 are trivial examples.
The first few numbers that are multiples of their reversals, excluding palindromic numbers and multiples
of 10, 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, copies of the same number 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 of these numbers, arranged in increasing order, 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 and Anderson 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)
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(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).
Examples of perfect powers with exponent greater than 2 resulting from reversal multiplication include
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(3)
|
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(4)
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