1971 Canadian MO Problems/Problem 7
Problem
Let be a five digit number (whose first digit is non-zero) and let
be the four digit number formed from n by removing its middle digit. Determine all
such that
is an integer.
Solution
Let and
, where
,
,
,
, and
are base-10 digits and
. If
is an integer, then
, or
This implies that
Clearly we have that , as
is positive. therefore
must be equal to 9, and
This simplifies to . The only way that this could happen is that
. Then
. Therefore the only values of
such that
is an integer are multiples of 1000.