2002 AMC 12A 第 17 题

Question

有若干组质数，例如

\{7, 83, 421, 659\}

，恰好各使用九个非零数字一次。这样的质数组的和最小可能是多少？

Several sets of prime numbers, such as

\{7, 83, 421, 659\},

use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?

193

207

225

252

477

数字

4, 6, 8

不能作为多位质数的个位，所以每个都必须是十位数字。

The digits

4, 6, 8

cannot end a prime past one digit, so each must be a tens digit

这三个数字至少贡献

40 + 60 + 80

，剩下六个数字至少按个位贡献它们本身的和；再检查是否存在可行例子。

Those three contribute at least

40 + 60 + 80,

and the remaining six digits at least their own sum as units digits; then check a valid set exists

偶数字

4, 6, 8

不能作为多位质数的个位，所以每个都必须出现在十位或更高位，至少贡献

40 + 60 + 80 = 180

。另外六个数字至少贡献

1 + 2 + 3 + 5 + 7 + 9 = 27

，因此总和至少为

207

。

这个下界可以达到，例如

\{2, 3, 5, 41, 67, 89\}

，它们的和为

207

。

因此，正确答案是 B。

The even digits

4, 6, 8

cannot be the units digit of a multi-digit prime, so each must appear in a tens place or higher, contributing at least

40 + 60 + 80 = 180.

The other six digits contribute at least

1 + 2 + 3 + 5 + 7 + 9 = 27,

so the sum is at least

207.

This bound is achieved, for example by

\{2, 3, 5, 41, 67, 89\},

whose sum is

207.

Thus, the correct answer is B.

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