AIME I 2025 (Problem 5)
Number TheoryMedium
There are eight-digit positive integers that use each of the digits exactly once. Let be the number of these integers that are divisible by . Find the difference between and .
If a number is divisible by , then it is divisible by and .
If a number is a multiple of , then the units digit is even.
If is a multiple of , then
Find the maximum and minimum of
By Hints 3 and 5,
By Hints 6 and 7,
so we should split to into two groups with equal sum , and each group should have elements.
Use casework based on the numbers grouped with .
( is the most limiting element.)
( is the most limiting element.)
Case 1:
There are cases so far.
There are cases so far.
WLOG,
should be even, so it has choices.
should be even, so it has choices.
The other cases are similar to Case 1.
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