2006 AMC 10B Problems/Problem 25
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 not the age of one of Mr. Jones's children?
Let be the set of the ages of Mr. Jones' children (in other words if Mr. Jones has a child who is years old). Then and . Let be the positive integer seen on the license plate. Since at least one of or is contained in , we have .
We would like to prove that , so for the sake of contradiction, let us suppose that . Then so the units digit of is . Since the number has two distinct digits, each appearing twice, another digit of must be . Since Mr. Jones can't be years old, the last two digits can't be . Therefore must be of the form , where is a digit. Since is divisible by , the sum of the digits of must be divisible by (see Divisibility rules for 9). Hence which implies . But is not divisible by , contradiction. So and is not the age of one of Mr. Jones' kids.
(We might like to check that there does, indeed, exist such a positive integer . If is not an age of one of Mr. Jones' kids, then the license plate number must be a multiple of . Since and is the only digit multiple of that fits all the conditions of the license plate's number, the license plate's number is .)
Alternatively, we can see that if one of Mr.Jones' children is of the age 5, then the license plate will have to end in the digit . This means that the license plate would have to have two digits, and would either be of the form or (X being the other digit in the license plate) . The condition is impossible as Mr. Jones can't be years old. If we separate the second condition, into its prime factors, we get . and are prime, therefore can't account for allowing being evenly divisible by the childrens' ages. This leaves , but no single digit number contains all the prime factors of the childrens ages, thus can't be one of the childrens ages.
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