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Number Theory

Source: 1998 National High School Mathematics League, Exam Two, Problem 3

March 9, 2020
number theory

Problem Statement

For positive integers a,na,n, define Fn(a)=q+rF_n(a)=q+r, where a=qn+ra=qn+r (q,rq,r are nonnegative integers, 0≤q<n0\leq q<n). Find the largest integer AA, there are positive integers n1,n2,n3,n4,n5,n6n_1,n_2,n_3,n_4,n_5,n_6, for all positive integer a≤Aa\leq A, Fn6(Fn5(Fn4(Fn3(Fn2(Fn1(a))))))=1F_{n_6}(F_{n_5}(F_{n_4}(F_{n_3}(F_{n_2}(F_{n_1}(a))))))=1.
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