+129899 Let p be the larger positive integer and q be the smaller
p + q = pq
Square this
p^2 + 2pq + q^2 = p^2q^2
p^2 + q^2 = p^2q^2 - 2pq (1)
And
p - q = 7
Square this and we get that
p^2 - 2pq + q^2 = 49
p^2 + q^2 = 49 + 2pq (2)
(1) and (2) imply that
p^2q^2 - 2pq = 49 + 2pq
p^2q^2 - 4pq = 49
Let pq = x
x^2 - 4x = 49 complete the square on x
x^2 - 4x + 4 = 49 + 4
(x - 2)^2 = 53 take both roots
x - 2 = √53 or x - 2 = -√53
x = 2 + √53 x = 2 - √53
pq = 2 + √53 pq = 2 - √53
We can simplify the given fraction as
p^2 q^2 (pq)^2
__________ = _________
p^2 + q^2 p^2 + q^2
We have two cases
pq = 2 + √53
(pq)^2 = 4 + 4√53 + 53 = 57 + 4√53
And p^2 + q^2 = 49 + 2(pq) = 49 + 2(2 + √53) = 53 + 2√53
So
(pq)^2 57 + 4√53 [ 53 - 2√53] 2597 + 98√53
_________ = ___________ ___________ = ______________
p^2 + q^2 53 + 2√53 [53 - 2√53] 2597
And
a = 2597, b = 98 , c = 53 and d = 2597 and their sum = 5345
Also
pq = 2 - √53
(pq)^2 = 4 - 4√53 + 53 = 57 - 4√53
And p^2 + q^2 = 49 + 2(pq) = 49 + 2(2 - √53) = 53 - 2√53
So
)pq)^2 57 - 4√53 [ 53 + 2√53 ] 2597 - 98√53
________ = ___________ ____________ = _____________
p^2 + q^2 53 - 2√53 [ 53 + 2√53] 2597
And
a = 2597 , b = -98 , c = 53 , d = 2597 and their sum = 5149
