AMC12 2002 A

AMC12 2002 A · Q17

AMC12 2002 A · Q17. It mainly tests Primes & prime factorization, Digit properties (sum of digits, divisibility tests).

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?

一些质数集合（例如 $\{7,83,421,659\}$）恰好把九个非零数字各用一次。这样的质数集合可能的最小和是多少？

(A) 193 193

(B) 207 207

(D) 252 252

(E) 477 477

Answer

Correct choice: (B)

正确答案：（B）

Solution

Neither of the digits $4$, $6$, and $8$ can be a units digit of a prime. Therefore the sum of the set is at least $40 + 60 + 80 + 1 + 2 + 3 + 5 + 7 + 9 = 207$. We can indeed create a set of primes with this sum, for example the following sets work: $\{ 41, 67, 89, 2, 3, 5 \}$ or $\{ 43, 61, 89, 2, 5, 7 \}$. Thus the answer is $207\implies \boxed{\mathrm{(B)}}$.

数字 $4$、$6$、$8$ 都不能作为质数的个位数字。因此该集合的和至少为 $40 + 60 + 80 + 1 + 2 + 3 + 5 + 7 + 9 = 207$。我们确实可以构造出和为 207 的质数集合，例如：$\{ 41, 67, 89, 2, 3, 5 \}$ 或 $\{ 43, 61, 89, 2, 5, 7 \}$。因此答案是 $207\implies \boxed{\mathrm{(B)}}$。

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