Question

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?

$4$

$5$

$6$

$7$

$8$

Solution:

Since a child is $9,$ the number is divisible by $9,$ so its digit sum $2(a+b)$ is a multiple of $9,$ which forces $a+b=9.$

There is also a $4$ - or $8$ -year-old, so the number is divisible by $4.$ Among numbers with two repeated digits summing to $9$ and divisible by $4,$ the number $5544$ is divisible by $1, 2, 3, 4, 6, 7, 8,$ and $9,$ but not by $5.$

So the eight ages can be $\{1,2,3,4,6,7,8,9\},$ and $5$ need not be among them. The age that is not necessarily a child's age is $5.$

Thus, the correct answer is B.

2006 AMC 10B Problem 25

Solution:

Problem 25 in Other Years