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MCIII Number Theory 2.26

 
 
JinTina的头像
MCIII Number Theory 2.26
JinTina - 2023年12月27日 Wednesday 17:50
 

Hello,


I tried considering parity for this one. After playing around with it for a while, I think it is impossible to make it all the same parity, so its impossible to make it the same number, but I can't prove it rigorously though. How to prove it rigorously?


Thanks,

Tina Jin

 
WangDr. Kevin的头像
Re: MCIII Number Theory 2.26
WangDr. Kevin - 2023年12月29日 Friday 22:32
 

Instead of using parity (which is mod 2), can you use another mod?  Especially, since each operation changes $3$ numbers by the same amount, the sum of all the numbers remains the same in mod 3. 

 
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JinTina的头像
Re: MCIII Number Theory 2.26
JinTina - 2024年01月4日 Thursday 21:58
 

I see, this is what I did:

Adding by any number does not change the sum mod 3 because it is adding by 3k. Summing everything mod 3, it is congruent to 1 mod 3, but for all numbers to be the same, it must be at-least a multiple of 3, contradiction, so it is impossible.

Thank you,

Tina Jin

 
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