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MCIII Number Theory 2.26

 
 
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MCIII Number Theory 2.26
by Tina Jin - Wednesday, December 27, 2023, 5:50 PM
 

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

 
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Re: MCIII Number Theory 2.26
by Dr. Kevin Wang - Friday, December 29, 2023, 10:32 PM
 

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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Picture of Tina Jin
Re: MCIII Number Theory 2.26
by Tina Jin - Thursday, January 4, 2024, 9:58 PM
 

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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