AMC8 2007

AMC8 2007 · Q19

AMC8 2007 · Q19. It mainly tests Linear equations, Factoring.

Pick two consecutive positive integers whose sum is less than 100. Square both of those integers and then find the difference of the squares. Which of the following could be the difference?

挑选两个连续的正整数，它们的和小于 100。将这两个整数平方，然后求平方差。以下哪一个可能是该差？

(A) 2 2

(B) 64 64

(D) 96 96

(E) 131 131

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) One of the squares of two consecutive integers is odd and the other is even, so their difference must be odd. This eliminates A, B and D. The largest consecutive integers that have a sum less than 100 are 49 and 50, whose squares are 2401 and 2500, with a difference of 99. Because the difference of the squares of consecutive positive integers increases as the integers increase, the difference cannot be 131. The difference between the squares of 40 and 39 is 79.

（C）两个相邻整数的平方，一个为奇数另一个为偶数，因此它们的差必为奇数。这排除了 A、B 和 D。和小于 100 的最大一对相邻整数是 49 和 50，它们的平方分别是 2401 和 2500，差为 99。由于相邻正整数平方的差会随着整数增大而增大，因此差不可能是 131。$40$ 与 $39$ 的平方之差是 79。

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