 Reducing both sides by multiples of 1024,
subtracting,
factoring,

The RHS must still be a multiple of 1024; since 53 is relatively prime to 1024, 5"F"+4 must be a multiple of 1024.The smallest such multiple is 11024, so 5"F"+4=1024 and F=204.Substituting into (1)

The Euclidean algorithm is quite tedious but a general methodology for solving rational equations ax+by=c requiring integral answers.