數學題目 急..........
有二個正整數相差60，且他們的正平方根之和又是某個正整數的平方根，若此平方根是一個無理數，是求所有滿足這些條件的任二個正整數！
1 則回答
最佳解答
不妨設二數為x – 30和x + 30其中x為正整數
因√(x – 30) + √(x + 30) = √y其中(x – 30), (x + 30)及y為正整數
[√(x – 30) + √(x + 30)]2 = y
2x + 2√[(x – 30)(x + 30)] = y
所以(x – 30)(x + 30)必為平方數,否則y不能為正整數
設(x – 30)(x + 30) = N2其中N為正整數
x2 – 900 = N2
(x – N)(x + N) = 900 並且 x + N > x – N
x + N = 較大因數…(1)
x – N = 較小因數…(2)
(1) + (2) => x = (較大因數 + 較小因數)/2
900的所有2因數分解如下:
900 = 1 * 900 => x = 450.5 (不合)
900 = 2 * 450 => x = 226 => √y = √196 + √256 = 30 (不合)
900 = 3 * 300 => x = 151.5 (不合)
900 = 4 * 225 => x = 114.5 (不合)
900 = 5 * 180 => x = 92.5 (不合)
900 = 6 * 150 => x = 78 => √y = √48 + √108 = 10√3 = √300
900 = 9 * 100 => x = 54.5 (不合)
900 = 10 * 90 => x = 50 => √y = √20 + √80 = 6√5 = √180
900 = 12 * 75 => x = 43.5 (不合)
900 = 15 * 60 => x = 37.5 (不合)
900 = 18 * 50 => x = 34 => √y = √4 + √64 = 10 (不合)
900 = 20 * 45 => x = 32.5 (不合)
900 = 25 * 36 => x = 30.5 (不合)
900 = 30 * 30 => x = 30 => (x – 30) = 0 (不合)
總結兩數為48及108或 20及80