Problem:
A telephone number has the form , where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, , and . Furthermore, , and are consecutive even digits; , and are consecutive odd digits; and . Find .
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The last four digits () are either or , and the remaining odd digit (either or ) is , , or . Since , the odd digit among , , and must be . Thus the sum of the two even digits in is . The three digits in are , , or , leaving the pairs and , and , or and , respectively, as the two even digits in . Of those, only the pair and has sum , so is , and the required first digit is . The only such telephone number is --.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions