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MCIII Number Theory 4.25

 
 
JinTina的头像
MCIII Number Theory 4.25
JinTina - 2024年01月1日 Monday 12:19
 

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

 
WangDr. Kevin的头像
Re: MCIII Number Theory 4.25
WangDr. Kevin - 2024年01月6日 Saturday 01:46
 

First consider: is $n$ even or odd?

 
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JinTina的头像
Re: MCIII Number Theory 4.25
JinTina - 2024年01月14日 Sunday 11:40
 

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.

 
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