Problem:
Positive integers x and y satisfy the equation x​+y​=1183​. What is the minimum possible value of x+y?
Answer Choices:
A. 585
B. 595
C. 623
D. 700
E. 791
Solution:
Observe that 1183=132⋅7, so 1183​=137​. Because x​+y​=137​, it follows that x​ and y​ must be of the form a7​ and b7​, respectively, where a and b are positive integers and a+b=13. Then x​=7a2​ and y​=7b2​, so x=7a2 and y=7b2. Substituting gives
x+y=7(a2+b2)=7(a2+(13−a)2)=14a2−182a+1183
The minimum value of the quadratic polynomial occurs at a=2⋅14182​=6.5. Because a and b must be positive integers, without loss of generality (by symmetry), choose a=6 and b=7. The minimum possible value of x+y is