Mock AIME 1 2006-2007/Problem 12
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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.
