Loser66
  • Loser66
How to prove a^3 -a divisible by 3 for all a? Please, help
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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SolomonZelman
  • SolomonZelman
Factors into: *a(a-1)(a+1)*
Loser66
  • Loser66
I know a^3 -a = a(a+1)(a-1) are 3 consecutive numbers, hence it divisible by3
SolomonZelman
  • SolomonZelman
so what's the prob/

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Loser66
  • Loser66
All I want is to go this way a is odd, a = 2k +1 a is even, a = 2k
Loser66
  • Loser66
but I got stuck at a is odd.
Loser66
  • Loser66
It is back to 3 consecutive numbers.
SolomonZelman
  • SolomonZelman
whether you start from odd or even integer, the product of 3 consequtive integers will be divisible by 3....
Loser66
  • Loser66
:) Hence, no way to avoid "3 consecutive numbers" method?? right?OOOOOOk, that's all I want to know. Thank you.
SolomonZelman
  • SolomonZelman
Yes, 3 conseq.... (or not another way that I aware of using my small knowledge)
freckles
  • freckles
you could do induction if you don't like that method above... \[\text{ assume } 3i=a^3-a \text{ for some integer } a \\ ... \\ (a+1)^3-(a+1)=a^3+3a^2+3a+1-a-1 \\ (a+1)^3-(a+1)=a^3-a+3a^2+3a \\ (a+1)^3-(a+1)=3i+3a^2+3a \\ (a+1)^3-(a+1)=3(i+a^2+a) \\ \text{ so } 3 \text{ is a factor of } (a+1)^3-(a+1)\]
freckles
  • freckles
and i is an integer of course
Loser66
  • Loser66
Got you. Thanks @freckles

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