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?

$4$

$5$

$6$

$7$

$8$

Answer: B

Concepts:divisibility digits casework

Difficulty rating: 1920

Solution:

The number has the form $aabb,$ $abab,$ or $baab.$ Divisibility by $9$ means $2(a + b)$ is a multiple of $9,$ so $a + b = 9.$

The eight distinct ages are eight of the nine integers from $1$ through $9,$ so at least one of ages $4$ and $8$ occurs. Therefore the number is divisible by $4.$ The possibilities become $1188, 2772, 3636,$ $5544, 6336, 7272, 9900.$

Since the last two digits are Mr. Jones's age, $9900$ is impossible, and none of the others is a multiple of $5.$ So the children's ages cannot include $5.$ Indeed $5544$ is divisible by $1, 2, 3, 4, 6, 7, 8, 9.$

Thus, the correct answer is B.

2006 AMC 12B Problem 19

Answer: B

Difficulty rating: 1920

Solution:

Problem 19 in Other Years