2. 外语
  3. The operationis defined for all integers x and y as xy = xy - y. If x and y are positive integers, which of the following CANNOT

The operationis defined for all integers x and y as xy = xy - y. If x and y are positive integers, which of the following CANNOT

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问题 The operationis defined for all integers x and y as xy = xy - y. If x and y are positive integers, which of the following CANNOT be zero?
选项 A、xy
B、yx
C、(x-l)y
D、(x+1)y
E、x(y- 1)
答案D
解析 In the formula xy = xy - y, the variables x and v are placeholders that can be replaced by integers or by expressions representing integers. Here are two examples.
    If x is replaced by 3 and y is replaced by 4, then the formula gives
                    34 =(3)(4)- 4 = 12-4 = 8
    If x is replaced by x - 1 and y is replaced by 2, then the formula gives
                (x- 1)2 =((x- l)(2))-2 = 2x-2-2 = 2x-4
Scanning the answer choices, you can see that all of them are of the form
                 "first expression""second expression"
For each answer choice, you must determine whether the answer choice can be equal to 0 for some positive integers x and y. Are there positive integers x and y for which the answer choice is equal to 0 ? If not, then that answer choice is the correct answer.
    Choice A: xy. Using the formula, try to find positive integers x and y for which xy = 0, that is, for which xy —y = 0. To solve this equation, note that factoring y out of the left-hand side of the equation xy - y = 0 gives the equation(x - l)y = 0. So now you must find positive integers x and y such that the product of the two numbers x -1 and y is 0. Since the product of two numbers is 0 only if at least one of the numbers is 0, it follows that the product of x - 1 and y will be 0 if x = 1, no matter what the value of y is. For example, if x = 1 and y = 2, then xy = 12 =(1)(2)- 2 = 0, and both x and y are positive integers. Therefore, Choice A is not correct, since there are positive integers x and y for which xy = 0.
    Choice B: yx. This is similar to Choice A, except the x and y are interchanged. Therefore, you might try the example in Choice A but with the values of x and y interchanged: y = 1 and x = 2. Using the formula, yx=yx-x =(1)(2)-2 = 0. Therefore, Choice B is not correct, since there are positive integers x and y for which yx = 0.
    Choice C:(x - l)y. Using the formula, try to find positive integers x and y for which(x - l)y = 0, that is, for which(x - l)y -y = 0. Factoring y out of the left-hand side of the equation(x - 1)y -y = 0 yields(x - 1 - l)y =(x - 2)y = 0. Here the product of the two numbers x - 2 and y is 0. So the product will be 0 if x = 2, no matter what the value of y is. For example, if x = 2 and y = 10, then(x- 1)y =(2- 1)10= 110 =(1)(10)- 10 = 0, and both x and y are positive integers. Therefore, Choice C is not correct, since there are positive integers x and y for which(x - 1)y = 0.
    Choice D:(x + 1)y. Using the formula, try to find positive integers x and y for which(x + 1)y = 0, that is, for which(x + 1)y — y - 0. Factoring y out of the left-hand side of the equation(x + 1)y —y = 0 yields(x + 1 - 1)y = xy = 0. Here the product of x and y is 0, so x = 0 or y = 0. Since both x and y must be positive but 0 is not positive, it follows that there are no positive integers x and y for which(x +1)y = 0. The correct answer is Choice D.
    Choice E: x(y - 1)cannot be correct, since Choice D is correct, but Choice E is considered here for completeness. Using the formula, try to find positive integers x and y for which x(y - 1)= 0, that is, for which x(y -1)—(y-l)= 0. Factoring y -1 out of the left-hand side of the equation x(y -1)—(y -1)= 0 yields(x - l)(y - 1)= 0. Here the product of the two numbers x - 1 and y -1 is 0. So the product will be 0 if x = 1 or y = 1, no matter what the value of the other variable is. For example, if x = 20 and y = 1, then x(y - 1)= 20(1 - 1)= 200 =(20)(0)-0 = 0, and both x and y are positive integers. Therefore, Choice E is not correct, since there are positive integers x and y for which x(y -1)= 0.
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