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Math Help - Sets without squares

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    Sets without squares

    Let S_i be the set of all integers n such that 100i\leq n < 100(i + 1). For example, S_4 is the set \{400,401,402,\ldots,499\}. How many of the sets S_0, S_1, S_2, \ldots, S_{999} do not contain a perfect square?
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    Quote Originally Posted by math sucks View Post
    Let S_i be the set of all integers n such that 100i\leq n < 100(i + 1). For example, S_4 is the set \{400,401,402,\ldots,499\}. How many of the sets S_0, S_1, S_2, \ldots, S_{999} do not contain a perfect square?
    For n≤50, the difference between n^2 and (n-1)^2 is less than 100. Since 50^2=2500, it follows that all the sets S_i for i≤25 will contain at least one square. For n>50, each square will differ from the previous one by more than 100, so each square will occupy a different S_i. The largest square less than 100,000 is 316^2=99,856. Thus the total number of sets S_i that do contain one or more squares is 26 (for the sets S_0 to S_25 inclusive) plus 266 (each containing exactly one of the squares from 51^2 to 316^2). Since 26+266=292, and there are altogether 1000 sets S_i (as i goes from 0 to 999), that leaves 1000292=708 sets that do not contain a perfect square.
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