In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3×3. Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number. A positive integer that has no perfect square divisors except 1 is called square-free . 1 By: Amit Sehgal Squares & Square roots 1 to 50

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A square number can end only with digits 0, 1 , 4, 6, 9, or 25 in base 10, as follows: 1.If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the digits preceding the ending 0s must also form a square. 2.If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four. 3.If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even. 4.If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four. 5.If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd. 6.If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56. 2 By: Amit Sehgal Squares & Square roots 1 to 50

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Odd and even square numbers Squares of even numbers are even (and in fact divisible by 4), since (2 n ) 2 = 4 n 2 . Squares of odd numbers are odd, since (2 n + 1) 2 = 4( n 2 + n ) + 1. It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd. As all even square numbers are divisible by 4, the even numbers of the form 4n + 2 are not square numbers. As all odd square numbers are of the form 4n + 1, the odd numbers of the form 4n + 3 are not square numbers. Squares of odd numbers are of the form 8n + 1, since (2 n + 1) 2 = 4 n ( n + 1) + 1 and n ( n + 1) is an even number. 3 By: Amit Sehgal Squares & Square roots 1 to 50

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4 By: Amit Sehgal 1 to 10 Square of 1 1 1.000 Square of 2 4 1.414 Square of 3 9 1.732 Square of 4 16 2.000 Square of 5 25 2.236 Square of 6 36 2.449 Square of 7 49 2.646 Square of 8 64 2.828 Square of 9 81 3.000 Square of 10 100 3.162 Squares & Square roots 1 to 50

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5 By: Amit Sehgal 11 to 20 Square of 11 121 3.317 Square of 12 144 3.464 Square of 13 169 3.606 Square of 14 196 3.742 Square of 15 225 3.873 Square of 16 256 4.000 Square of 17 289 4.123 Square of 18 324 4.243 Square of 19 361 4.359 Square of 20 400 4.472 Squares & Square roots 1 to 50

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6 By: Amit Sehgal 21 to 30 Squares & Square roots 1 to 50 Square of 21 441 4.583 Square of 22 484 4.690 Square of 23 529 4.796 Square of 24 576 4.899 Square of 25 625 5.000 Square of 26 676 5.099 Square of 27 729 5.196 Square of 28 784 5.292 Square of 29 841 5.385 Square of 30 900 5.477

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7 By: Amit Sehgal 31 to 40 Squares & Square roots 1 to 50 Square of 31 961 5.568 Square of 32 1,024 5.657 Square of 33 1,089 5.745 Square of 34 1,156 5.831 Square of 35 1,225 5.916 Square of 36 1,296 6.000 Square of 37 1,369 6.083 Square of 38 1,444 6.164 Square of 39 1,521 6.245 Square of 40 1,600 6.325

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8 By: Amit Sehgal 41 to 50 Squares & Square roots 1 to 50 Square of 41 1,681 6.403 Square of 42 1,764 6.481 Square of 43 1,849 6.557 Square of 44 1,936 6.633 Square of 45 2,025 6.708 Square of 46 2,116 6.782 Square of 47 2,209 6.856 Square of 48 2,304 6.928 Square of 49 2,401 7.000 Square of 50 2,500 7.071

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