2. 外语
  3. Let n and k be positive integers with k< n. From an n x n array of dots, a k x k array of dots is selected. The figure above sho

Let n and k be positive integers with k< n. From an n x n array of dots, a k x k array of dots is selected. The figure above sho

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问题
Let n and k be positive integers with k< n. From an n x n array of dots, a k x k array of dots is selected. The figure above shows two examples where the selected k x k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n x n array are NOT in the selected k x k array?
选项 A、1
B、2
C、3
D、4
E、5
答案C
解析 The n × n array has n2 dots and the k × k array has k2 dots. The number of dots in the n × n array that are not in the k x k array is given by n2-k2 = (n-k)(n + k).
Therefore, (n - k)(n + k) - 48 is a necessary condition for there to be 48 dots not in the k × k array. This is also a sufficient condition, since it is clear that at least one k x k array of dots can be selected for removal from an× n array of dots when k ≤ n.
The equation (n - k)(n + k) = 48 represents two positive integers, namely n - k and n + k，whose product is 48. Thus, the smaller integer n - k must be 1,2,3,4, or 6, and the larger integer n + k must be 48,24,16,12, or 8. Rather than solving five pairs of simultaneous equations (for example, n - k = 2 and n + k = 24 is one such pair), it is more efficient to observe that the solution to the
system n - k = a and n + k = b is n = = (a + b)/2 (add the equations, then divide by 2) and k = - (b - a)/2(substitute n = (a + b)/2 for n in either equation and solve for k; or subtract the equations, then divide by 2). Therefore, the possible pairs (n, k) arise exactly when 48 = ab and both a + b and b - a are divisible by 2. This occurs exactly three times—48 = (2)(24), 48 = (4)(12), and 48 = (6)(8).
The correct answer is C.
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