Problem:
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9 , spots a license plate with a 4 -digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is the age of one of Mr. Jones's children?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The 4-digit number on the license plate has the form or or , where and are distinct integers from 0 to 9 . Because Mr. Jones has a child of age 9 , the number on the license plate is divisible by 9 . Hence the sum of the digits, , is also divisible by 9 . Because of the restriction on the digits and , this implies that . Moreover, since Mr. Jones must have either a 4 -year-old or an 8 -year-old, the license plate number is divisible by 4 . These conditions narrow the possibilities for the number to , 6336, 7272, and 9900. The last two digits of 9900 could not yield Mr. Jones's age, and none of the others is divisible by 5 , so he does not have a .
Note that 5544 is divisible by each of the other eight non-zero digits.
The problems on this page are the property of the MAA's American Mathematics Competitions