Divisor game

Number Theory Level 4

Let n > 1 n>1 be a positive integer. Sharky and Ivan are playing an interesting game, with Sharky going first. They take turns to write exactly one positive divisor of n n onto a board. The next player in sequence can't write a number which is a divisor of a number already written. The player who writes n n onto the board loses.

For how many n n does Ivan have a winning strategy?

0 1 2 3 4 5 Infinitely many

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Mike Davison
Oct 17, 2017

I'm looking forward to seeing an official solution...

As far as I can tell, if n has exactly two prime factors eg "10", Sharky can't win? Or does Sharky choose n?

2 Helpful 0 Interesting 0 Brilliant 0 Confused

For every n, where n is prime only sharky can win. For n, where it has even no. of prime divisors only, then Ivan can always have a winning strategy if sharky starts with a prime factor. Such n which have even no. of prime factors only is infinite.

Fake Human - 3 years, 6 months ago

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Yah!!! I agree.

Chan Tin Ping - 3 years, 5 months ago

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