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number theory senior
Problem
The number is to be written with a factorial base of numeration, that is, where are integers such that and means . Find
(A)
(B)
(C)
(D)
Solution
This problem can be approached similarly to other base number problems. Since , divide by . The quotient is and the remainder is , so rewrite the number as Similarly, dividing by results in a quotient of and a remainder of , so the number can be rewritten as Repeat the steps to get The answer is . One can also stop at the second step by noting .
Final answer
D