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Miscellaneous Elementary Junior High Sum Of Positive Integral Divisors High School Manipulating Polygonal Numbers
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Working With Square Roots:A. The trick to working with square roots is to know what range the square root is in. Since squaring numbers ending in 0 and squaring numbers ending in 5 are both easy, we can get the answer within a range of 5. B. If the number is a perfect square, then we can know what the ending digit will be of the answer by looking at the ending digit of the question. If the number ends in a: 0 -> then the ending digit is a 0 1 -> then the ending digit is 1 or 9. 4 -> then the ending digit is 2 or 8. 5 -> then the ending digit is a 5. 6 -> then the ending digit is 4 or 6. 9 -> then the ending digit is 3 or 7. C. After finding the last digit (or possibility between two digits) mentally chop off the last two digits and focus on the remaining digits. D. Now, try to find a range of 5 that the number (in step c) is in. Once you do this you know the answer using step B. First, find a range of 10, then find out if the answer is in the upper-half of the range (i.e. ends in 5, 6, 7, 8, or 9) or if the answer is in the lower-half of the range (i.e. ends in 0, 1, 2, 3, or 4) by squaring the middle number (the number in the range that ends in 5). Ex [1] /(5329) = _________. a. We know the last number must be a 3 or 7. b. Chopping off the last two numbers we need to focus on the remaining numbers or 53. c. We know that 72 = 49 and 82 = 64 so since 53 is between these two numbers, the answer is between 70 and 80. d. Squaring 75 we get 5625. Since the original number is lower than this, we know the answer is in the lower-half. e. The answer is 73. Ex [2] /(15876) = __________. a. We know that the last number must be a 4 or 6. b. Chopping off the last two numbers we are left with 158. c. We know that 122 = 144 and 132 = 169 so the number is between 120 and 130 since 144 < 158 < 169. d. Squaring 125, we get 15625. Since the original number is greater than this, we know the answer is in the upper-half. e. The answer is 126.
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