2006 AMC 12B Problem 19

Problem 19 of 25HarderNumber Theory

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?

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Solution

The number has the form

aabb,

abab,

baab.

Divisibility by

9

means

2(a + b)

is a multiple of

9,

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.

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