Online Course Discussion Forum

MCIII Number Theory 4.25

 
 
Picture of Tina Jin
MCIII Number Theory 4.25
by Tina Jin - Monday, 1 January 2024, 12:19 PM
 

Hello,

Hello! For this problem, I randomly guessed the sequence:

a1=-4

a2=-2

a3=1

a4=1

a5=1

a6=1

a7=1

a8=1

So a1+a2+...+a8=0, and a1*a2*...*a8=8, so it satisfies and n congruent to 0 (mod 4).

Is there are more rigorous way to do this?

Thanks,

Tina Jin

 
Picture of Dr. Kevin Wang
Re: MCIII Number Theory 4.25
by Dr. Kevin Wang - Saturday, 6 January 2024, 1:46 AM
 

First consider: is $n$ even or odd?

Picture of Tina Jin
Re: MCIII Number Theory 4.25
by Tina Jin - Sunday, 14 January 2024, 11:40 AM
 

I understand now! When n=odd, the sum of ai cannot be even number (namely cannot be 0). When n=even, there must be more than 1 even ai, so n is a multiple of 4 and therefor congruent to 0 mod 4.