2025 AMC 8 第 23 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
23.
有多少个四位数同时满足以下三个性质？
(I) 十位数字和个位数字都是 。
(II) 这个数比一个完全平方数小 。
(III) 这个数是恰好两个质数的乘积。
How many four-digit numbers have all three of the following properties?
(I) The tens digit and ones digit are both
(II) The number is less than a perfect square.
(III) The number is the product of exactly two prime numbers.
小提示：
比它大一的平方数必须以 结尾
The square one larger must end in
大提示：
检查哪些 的倍数两边相邻数都是质数
Check multiples of whose two neighbors are prime
视频讲解：
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文字解答：
这个数形如 ，所以大 的完全平方数以 结尾，因此它是一个以 结尾的数的平方。设这个平方为 。
为了让 是一个 位数且末两位为 ， 只能是 。
由于 ，还需要 和 都是质数。也就是说，要在 附近寻找相邻的质数对。
逐一检查，只有 和 可行，因此这样的数恰有 个。答案是 B。
The number has the form and so the perfect square that is more ends in and so it is the square of a number ending in Suppose that square is
In order for to be a -digit number ending in , the only possibilities for are
Since we also need both and to be prime. We are then looking for pairs of prime numbers that are right around
Going through all the possibilities, the only ones that work are and , and so there is exactly way to do this. The answer is B.
其他年份的第 23 题
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