AMC8 1997

AMC8 1997 · Q23

AMC8 1997 · Q23. It mainly tests Primes & prime factorization, Digit properties (sum of digits, divisibility tests).

There are positive integers that have these properties: I. The sum of the squares of their digits is 50, and II. Each digit is larger than the one on its left. The product of the digits of the largest integer with both properties is

存在具有以下性质的正整数：I. 其各位数字平方和为 50，且 II. 每位数字都大于其左边的数字。具有两者性质的最大整数的各位数字乘积是

(A) 7 7

(B) 25 25

(D) 48 48

(E) 60 60

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): To meet the first condition, numbers which sum to 50 must be chosen from the set of squares $\{1,4,9,16,25,36,49\}$. To meet the second condition, the squares selected must be different. Consequently, there are three possibilities: $1+49$, $1+4+9+36$, and $9+16+25$. These correspond to the integers 17, 1236, and 345, respectively. The largest is 1236, and the product of its digits is $1\cdot 2\cdot 3\cdot 6=36$.

答案（C）：为满足第一个条件，必须从平方数集合 $\{1,4,9,16,25,36,49\}$ 中选出和为 50 的数。为满足第二个条件，所选的平方数必须互不相同。因此，有三种可能：$1+49$、$1+4+9+36$、以及 $9+16+25$。它们分别对应整数 17、1236 和 345。最大的是 1236，其各位数字的乘积为 $1\cdot 2\cdot 3\cdot 6=36$。

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