AMC 12B 2025 (Problem 5)Positive integers xxx and yyy satisfy the equation 57x+22y=40057x+22y=40057x+22y=400. What is the least possible value of x+yx+yx+y?(A) 10\text{(A)}\;10(A)10(B) 11\text{(B)}\;11(B)11(C) 13\text{(C)}\;13(C)13(D) 14\text{(D)}\;14(D)14(E) 15\text{(E)}\;15(E)15Related TopicsCoreToolkit 111 — Linear Diophantine EquationsHints (4)Hint 1Let x+y=sx+y=sx+y=s.57x+22y=40057x+22y=40057x+22y=40035x+22(x+y)=40035x+22(x+y)=40035x+22(x+y)=40035x+22s=40035x+22s=40035x+22s=400Hint 2xxx and sss must be multiples of 222 and 555, respectively.Hint 3Let x=2x′x=2x'x=2x′ and s=5s′s=5s's=5s′.35(2x′)+22(5s′)=40035(2x')+22(5s')=40035(2x′)+22(5s′)=4007x′+11s′=407x'+11s'=407x′+11s′=40min:s′=3,x′=1\min:\quad s'=3,\quad x'=1min:s′=3,x′=1Hint 4s=5s′=15,x=2x′=2,y=13 ✓s=5s'=15,\quad x=2x'=2,\quad y=13\;\checkmarks=5s′=15,x=2x′=2,y=13✓Final Answer(E) 151515Related Problems (2)AMC 10B/12B 2025 (Problem 23/19)AMC 10B 2025 (Problem 25)