商與差相同的兩個數字

設兩數為 a , b :a - b = a/bab - b^2 = ab^2 - ab + a = 0b = [a ± √ (a^2 - 4a)] / 2因 b 為整數 , 故 √ (a^2 - 4a) 是整數 , 即 a^2 - 4a 是平方數 ,設
a^2 - 4a = k^2 ..............(*)a^2 - 4a - k^2 = 0a = { 4 ± √[(4^2 - 4(-k^2)] } / 2a = 2 ± √ (4 + k^2)因 a 為整數 , 故 √ (4 + k^2) 是整數 , 即 4 + k^2 是平方數 ,設 4 + k^2 = p^2 ,p^2 - k^2 = 4(p - k)(p + k) = 4當 p - k = ±1 , p + k = ±4 ==> p = ±2.5 , k = ±1.5 (不是整數不合) ;當 p - k = ±2 , p + k = ±2==> p = ±2 , k = 0 代 k = 0 入 (*) :a^2 - 4a = 0a(a - 4) = 0a = 0 或 a = 4 ,b = 0(捨去因 0/0 無意義) 或 b = 2

只有 4 和 2

a - b= a/b
ab - b^2 = a
a = b^2 / (b - 1) = b + 1 + 1/(b-1) => b = 2; a=4

考慮 a - b = a/b
設 a/b = k, 則 a = bk
bk - b = k
b = k/(k - 1) =(k - 1 + 1)/(k - 1) = 1 + 1/(k - 1)
1/(k - 1) = b - 1

k - 1 整除 1
k - 1 = 1 或 -1(捨去)
k = 2

b = 2/(2 - 1) = 2
a = 2(2) = 4