AMC12 2006 B

AMC12 2006 B · Q19

AMC12 2006 B · Q19. It mainly tests Divisibility & factors, GCD & LCM.

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?

琼斯先生有八个不同年龄的孩子。在一次家庭旅行中，他最大的孩子（9 岁）看到一个车牌号是一个 4 位数，其中恰有两个数字各出现两次。“看，爸爸！”她喊道，“这个数能被我们每个孩子的年龄整除！”“没错，”琼斯先生回答，“而且最后两位数字恰好是我的年龄。”以下哪项不是琼斯先生某个孩子的年龄？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (B)

正确答案：（B）

Solution

First, The number of the plate is divisible by $9$ and in the form of $aabb$, $abba$ or $abab$. We can conclude straight away that $a+b= 9$ using the $9$ divisibility rule. If $b=1$, the number is not divisible by $2$ (unless it's $1818$, which is not divisible by $4$), which means there are no $2$, $4$, $6$, or $8$ year olds on the car, but that can't be true, as that would mean there are less than $8$ kids on the car. If $b=2$, then the only possible number is $7272$. $7272$ is divisible by $4$, $6$, and $8$, but not by $5$ and $7$, so that doesn't work. If $b=3$, then the only number is $6336$, also not divisible by $5$ or $7$. If $b=4$, the only number is $5544$. It is divisible by $4$, $6$, $7$, and $8$. Therefore, we conclude that the answer is $\mathrm{(B)}\ 5$ NOTE: Automatically, since there are 8 children and all of their ages are less than or equal to 9 and are different, the answer choices can be narrowed down to $5$ or $8$.

首先，车牌号码能被 $9$ 整除，且形式为 $aabb$、$abba$ 或 $abab$。由 $9$ 的整除规则可立刻推出 $a+b= 9$。若 $b=1$，则该数不能被 $2$ 整除（除非是 $1818$，但它不能被 $4$ 整除），这意味着车上没有 $2$、$4$、$6$ 或 $8$ 岁的孩子，但这不可能，因为那样车上的孩子数会少于 $8$。若 $b=2$，则唯一可能的数是 $7272$。$7272$ 能被 $4$、$6$ 和 $8$ 整除，但不能被 $5$ 和 $7$ 整除，所以不行。若 $b=3$，则唯一的数是 $6336$，同样不能被 $5$ 或 $7$ 整除。若 $b=4$，唯一的数是 $5544$。它能被 $4$、$6$、$7$ 和 $8$ 整除。因此可得答案为 $\mathrm{(B)}\ 5$ 注：由于有 8 个孩子且年龄互不相同并且都不超过 9，选项可直接缩小到 $5$ 或 $8$。

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