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game on sequence

Source: 9th EMC, 12th December 2020 - 20th December 2020. SENIOR league, P3.

December 22, 2020
number theorygameprime numbers

Problem Statement

Let pp be a prime number. Troy and Abed are playing a game. Troy writes a positive integer XX on the board, and gives a sequence (an)n∈N(a_n)_{n\in\mathbb{N}} of positive integers to Abed. Abed now makes a sequence of moves. The nn-th move is the following:  Replace Y currently written on the board with either Y+an or Y⋅an.\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n. Abed wins if at some point the number on the board is a multiple of pp. Determine whether Abed can win, regardless of Troy’s choices, if a)p=109+7a) p = 10^9 + 7; b)p=109+9b) p = 10^9 + 9. Remark: Both 109+710^9 + 7 and 109+910^9 + 9 are prime.
Proposed by Ivan Novak
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