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四个整数

Four Integers

专题
Discrete Math / 离散数学
难度
L2

题目详情

我们有 4 个正整数 A、B、C 和 D。我们知道 AB = 16、BC = 14 和 CD = 63。A + B + C + D 是多少？

We have 4 positive integers A, B, C, and D. We know AB = 16, BC = 14, and CD = 63. What is A + B + C + D?

解析

由于 AB = 16，我们知道 A 和 B 一定是 16 的因数。 16的正因数为1、2、4、8、16。因此，A和B只能取(8,2)、(2,8)或(4,4)。

可能的组合

  1. A*B = 4 * 4，则 A = 4 且 B = 4
  2. A*B = 8 * 2 则 A = 8 且 B = 2
  3. A*B = 2 * 8 则 A = 2 且 B = 8

由于 BC = 14，我们知道 B 和 C 一定是 14 的因数。 14的正因数是1、2、7和14。之前我们发现B只能是2、4或8。由于2是B唯一可能出现的14的因数，所以B一定是2，C一定是7。此外，由于B是2，我们也可以得出A一定是8的结论。

最后，我们有 CD = 63。因为我们知道 C 一定是 7，所以我们可以解出等于 9 的 D。

因此，A + B + C + D = 8 + 2 + 7 + 9 = 26

Original Explanation

Since AB = 16, we know that A and B must be factors of 16. The positive factors of 16 are 1, 2, 4, 8, and 16. Therefore, A and B can only take on the values (8, 2), (2, 8) or (4, 4).

Possible Combinations

  1. A*B = 4 * 4, then A = 4 and B = 4
  2. A*B = 8 * 2 then A = 8 and B = 2
  3. A*B = 2 * 8 then A = 2 and B = 8

Since BC = 14, we know that B and C must be factors of 14. The positive factors of 14 are 1, 2, 7, and 14. Previously, we found that B can only be 2, 4 or 8. Since 2 is the only factor of 14 that B can possibly be, B must be 2 and C must be 7. Furthermore, since B is 2, we can also conclude that A must be 8.

Finally, we have CD = 63. Since we know that C must be 7, we can solve for D which equates to 9.

Therefore, A + B + C + D = 8 + 2 + 7 + 9 = 26