MATHCOUNTS State Sprint 2025 (Problem 29)Number TheoryMediumLet SSS be the sum of all the positive integers that are factors of 999,999999,999999,999. What is the largest prime number that is a factor of SSS?Related TopicsCoreDivisor FunctionsCheck AnswerYour answer:CheckHints (3)Hint 1Prime factorize 999999999999999999.Hint 2999999=33⋅7⋅11⋅13⋅37999999=3^3\cdot7\cdot11\cdot13\cdot37999999=33⋅7⋅11⋅13⋅37Hint 3By Sum of Positive Divisors,S=(1+3+32+33)(1+7)(1+11)(1+13)(1+37)S=(1+3+3^2+3^3)(1+7)(1+11)(1+13)(1+37)S=(1+3+32+33)(1+7)(1+11)(1+13)(1+37)=40⋅8⋅12⋅14⋅38=40\cdot8\cdot12\cdot14\cdot38=40⋅8⋅12⋅14⋅38=23⋅5⋅23⋅22⋅3⋅2⋅7⋅2⋅19=2^3\cdot5\cdot2^3\cdot2^2\cdot3\cdot2\cdot7\cdot2\cdot19=23⋅5⋅23⋅22⋅3⋅2⋅7⋅2⋅19Final Answer191919Related Problems (1)AMC 12B 2025 (Problem 23)