75 arithmetic/digits/reverse.p
Description
This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.
75 arithmetic/digits/reverse.p
Is there an integer that has its digits reversed after dividing it by 2?
arithmetic/digits/reverse.s
Assume there's such a positive integer x such that x/2=y and y is the
reverse of x.
Then x=2y. Let x = a...b, then y = b...a, and:
b...a (y)
x 2
--------
a...b (x)
From the last digit b of x, we have b = 2a (mod 10), the possible
values for b are 2, 4, 6, 8 and hence possible values for (a, b) are
(1,2), (6,2), (2,4), (7,4), (3,6), (8,6), (4,8), (9,8).
From the first digit a of x, we have a = 2b or a = 2b+1. None of the
above pairs satisfy this condition. A contradiction.
Hence there's no such integer.
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