Article 278731 of rec.puzzles From: Dan Hoey Newsgroups: rec.puzzles Subject: Re: cotpi 17 - Product Date: Mon, 22 Aug 2011 16:36:29 -0400 Organization: Naval Research Laboratory, Washington, DC cotpi wrote: > There is a list of 7 unique positive integers less than 10. One can > not deduce whether their sum is a multiple of 3 from their product. > What is their product? The word "unique" is makes no sense here. I assume you mean "distinct." It is easier to solve this problem by considering the two positive integers from {1,2,3,...9} that are not among the seven given. The product of the two numbers is in one-to-one correspondence with the product of the seven given, and the sum of the two numbers mod 3 is the negative of the sum of the sum of the seven given. Therefore, a necessary and sufficient condition is one cannot deduce whether the sum of the two is a multiple of three from their product. If the product were 1 mod 3, then the two numbers would either both be 1 mod 3 or both be 2 mod 3, so one could deduce that the sum is not a multiple of 3. If the product were 2 mod 3, then the two numbers would be 1 mod 3 and two mod 3, so one could deduce that the sum is a multiple of 3. Therefore, the product is divisible by 3. If the product were not divisible by 9, then one of the numbers would be a multiple of 3 and the other a non-multiple of 3, so one could deduce that the sum is not a multiple of 3. Therefore the product is divisible by 9, and we can't tell from their product whether the two numbers are {x,9y} or {3x,3y}. In order that the numbers be less than 10 and distinct, x=2 and y=1. Therefore, the seven given integers are either {1,3,4,5,6,7,8} or {1,2,4,5,7,8,9} and their product is 20160. Dan