2. 外语
  3. The integer v is greater than 1. If v is the square of an integer, which of the following numbers must also be the square of an

The integer v is greater than 1. If v is the square of an integer, which of the following numbers must also be the square of an

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问题 The integer v is greater than 1. If v is the square of an integer, which of the following numbers must also be the square of an integer?
Indicate all such numbers.
选项 A、 
B、 
C、 
答案A，B
解析 If v is the square of an integer, then is an integer. You can use this fact, together with the fact that the product and the sum of integers are also integers, to examine the first two choices.
    Choice A: The positive square root of 81 v is 9, which is an integer. So 81v is the square of an integer.
    Choice B: 25v +10+ 1 =(5+ l)2 and 5+ 1 is an integer. So 25v + 10+ 1 is the square of an integer.
    Choice C: Since there is no obvious way to factor the given expression, you may suspect that it is not the square of an integer. To show that a given statement is not true, it is sufficient to find one counterexample. In this case, you need to find one value of v such that v is the square of an integer but 4v2 + 4+ 1 is not the square of an integer. If v = 4, then 4v2 + 4+ 1 = 64 + 8 + 1 = 73, which is not the square of an integer. This proves that 4v2 + 4+ 1 does not have to be the square of an integer.
    The correct answer consists of Choices A and B.
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