AMC 10A 2025 (Problem 17)Let NNN be the unique positive integer such that dividing 273436273436273436 by NNN leaves a remainder of 161616 and dividing 272760272760272760 by NNN leaves a remainder of 151515. What is the tens digit of NNN?(A) 0\text{(A)}\;0(A)0(B) 1\text{(B)}\;1(B)1(C) 2\text{(C)}\;2(C)2(D) 3\text{(D)}\;3(D)3(E) 4\text{(E)}\;4(E)4Related TopicsCoreToolkit 79 — Computing gcd(a,b)MinorToolkit 47 — Modular Arithmetic: Definition and PropertiesToolkit 46 — Divisibility: Definition and PropertiesHints (7)Hint 1273436≡16(modN) ⇒ N∣273420273436\equiv16\pmod N\ \Rightarrow\ N\mid273420273436≡16(modN) ⇒ N∣273420Hint 2272760≡15(modN) ⇒ N∣272745272760\equiv15\pmod N\ \Rightarrow\ N\mid272745272760≡15(modN) ⇒ N∣272745Hint 3If a∣ba\mid ba∣b and a∣ca\mid ca∣c, then a∣gcd(b,c)a\mid\gcd(b,c)a∣gcd(b,c).Hint 4N∣gcd(273420,272745)N\mid\gcd(273420,272745)N∣gcd(273420,272745)Hint 5If a>ba>ba>b:gcd(a,b)=gcd(remainder of (a÷b),b)\gcd(a,b)=\gcd(\text{remainder of }(a\div b),b)gcd(a,b)=gcd(remainder of (a÷b),b)Hint 6gcd(273420,272745)=gcd(675,272745)\gcd(273420,272745)=\gcd(675,272745)gcd(273420,272745)=gcd(675,272745)=gcd(675,45)=45=\gcd(675,45)=45=gcd(675,45)=45Hint 7Since N>remainder=16N>\text{remainder}=16N>remainder=16 and N∣45N\mid45N∣45,N=45N=45N=45Final Answer(E) 444Related Problems (1)AMC 12A 2022 (Problem 16)