2024 AIME II Problems/Problem 7
Problem
Let be the greatest four-digit positive integer with the property that whenever one of its digits is changed to
, the resulting number is divisible by
. Let
and
be the quotient and remainder, respectively, when
is divided by
. Find
.
Solution 1
We note that by changing a digit to for the number
, we are subtracting the number by either
,
,
, or
. Thus,
. We can casework on
backwards, finding the maximum value.
(Note that computing greatly simplifies computation).
Applying casework on , we can eventually obtain a working value of
. ~akliu
Solution 2
Let our four digit number be . Replacing digits with 1, we get the following equations:
Reducing, we get
Subtracting , we get:
For the largest 4 digit number, we test values for a starting with 9. When a is 9, b is 4, c is 3, and d is 7. However, when switching the digits with 1, we quickly notice this doesnt work. Once we get to a=5, we get b=6,c=9,and d=4. Adding 694 with 5, we get -westwoodmonster