Running the Recursion Backwards

★ 6/10number theory Recursion Modular Arithmetic sequences Casework

Problem

A sequence satisfies

a_{n+1} = \left\lfloor\frac{a_n}{2}\right\rfloor + \left\lfloor\frac{a_n}{3}\right\rfloor, \quad n = 1, 2, \dots, 7,

where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$ . If $a_8 = 8$ , how many possible values can the positive integer $a_1$ take?

Answer

Solution

Difficulty	6/10
Topics	number theory, Recursion, Modular Arithmetic, sequences, Casework

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