Problem:
There are four-digit whole numbers that use each of the four digits and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Six of the numbers are in the , six in the , six in the and six in the . Doubling and tripling numbers in the produce possible solutions, but any multiple of those in the other sets is larger than .
Units digits of the numbers are and , so their doubles will end in and , respectively. Choice ends in but , not one of the numbers. Likewise, choice produces , also not one of the numbers. When the units digits are tripled the resulting units digits are and and choices , and are possibilities. Division by yields and respectively. Only the second of these numbers is one of the given numbers. Choice is correct.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions