# 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.