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 (GHIJ) are either 9753 or 7531 , and the remaining odd digit (either 1 or 9 ) is , or . Since , the odd digit among , and must be 1. Thus the sum of the two even digits in is 8 . The three digits in DEF are 864,642 , or 420 , leaving the pairs 2 and 0,8 and 0 , or 8 and 6 , respectively, as the two even digits in ABC. Of those, only the pair 8 and 0 has sum 8 , so is 810 , and the required first digit is . The only such telephone number is .
The problems on this page are the property of the MAA's American Mathematics Competitions