Mock AIME 1 2006-2007 Problems/Problem 12
Problem
Let be a positive integer with first digit 4 such that after removing the first digit, you get another positive integer,
, that satisfies
. Find the number of possible values of
between
and
.
Solution
The digit-removal condition is equivalent to the statement where
and
. Thus
so
and
. It's easy to see that this value of
is small enough, so all we need to check is that it is an integer. That happens if and only if 13 is a divisor of
, so
and multiplying by
we have that
Certainly
is a solution. All we need is the order of 10
. Now
so
,
and the order of 10 mod 13 is 6. Thus, we get one value of
each time
. There are
such values of
which fall in the required range.