Problem:
How many -digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of , and is the largest digit?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Any integer that is a multiple of must have a or as the units digit.
If the units digit is , then the other three digits must be a and two digits selected from , and . After the pair is selected, there are possible ways to arrange them to form the number. So there are possible numbers with units digit .
If the units digit is , then there are four choices for the thousands digit and there are ways to complete the number. So there are possible numbers with units digit .
Together there are possible numbers.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions