AMC 10A/12A 2024 (Problem 18/11)There are exactly KKK positive integers 5≤b≤20245\le b\le20245≤b≤2024 such that the base-bbb integer 2024b2024_b2024b is divisible by 161616 (where 161616 is in base ten). What is the sum of the digits of KKK?(A) 16\text{(A)}\;16(A)16(B) 17\text{(B)}\;17(B)17(C) 18\text{(C)}\;18(C)18(D) 20\text{(D)}\;20(D)20(E) 21\text{(E)}\;21(E)21Related TopicsCoreNumber BasesModular ArithmeticMinorProperties of InequalitiesFloor and Ceiling FunctionsCheck AnswerYour answer:ABCDECheckHints (10)Hint 12024b=4+2b+0b2+2b32024_b=4+2b+0b^2+2b^32024b=4+2b+0b2+2b3Hint 22b3+2b+4≡0(mod16)2b^3+2b+4\equiv0\pmod{16}2b3+2b+4≡0(mod16)Hint 3b3+b+2≡0(mod8)b^3+b+2\equiv0\pmod{8}b3+b+2≡0(mod8)Hint 4b≡0,1,−1,2,−2,3,−3,4(mod8)b\equiv0,1,-1,2,-2,3,-3,4\pmod{8}b≡0,1,−1,2,−2,3,−3,4(mod8)Hint 5b3+b+2≡2×,4×,0✓,4×,0✓,0✓,4×,6×(mod8)b^3+b+2\equiv\underset{\times}{2},\underset{\times}{4},\underset{\checkmark}{0},\underset{\times}{4},\underset{\checkmark}{0},\underset{\checkmark}{0},\underset{\times}{4},\underset{\times}{6}\pmod{8}b3+b+2≡×2,×4,✓0,×4,✓0,✓0,×4,×6(mod8)Hint 6⇒b≡−1,−2,3≡7,6,3(mod8)\Rightarrow b\equiv-1,-2,3\equiv7,6,3\pmod{8}⇒b≡−1,−2,3≡7,6,3(mod8)Hint 7b≡3(mod8)⇒20248−1=253−1=252b\equiv3\pmod{8}\Rightarrow\frac{2024}{8}-1=253-1=252b≡3(mod8)⇒82024−1=253−1=252Hint 8b≡6(mod8)⇒20248=253b\equiv6\pmod{8}\Rightarrow\frac{2024}{8}=253b≡6(mod8)⇒82024=253Hint 9b≡7(mod8)⇒20248=253b\equiv7\pmod{8}\Rightarrow\frac{2024}{8}=253b≡7(mod8)⇒82024=253Hint 10252+253+253=758⇒Ans=7+5+8=20252+253+253=758\Rightarrow\text{Ans}=7+5+8=20252+253+253=758⇒Ans=7+5+8=20Related Problems (8)AMC 10B Fall 2021 (Problem 22)AIME I 2026 (Problem 8)AMC 10B/12B 2025 (Problem 4/4)AIME II 2026 (Problem 4)AMC 10B/12B 2025 (Problem 8/6)AMC 12B 2025 (Problem 9)AMC 12B 2025 (Problem 23)AIME II 2026 (Problem 3)Final Answer(D) 20