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All primes except 3 are congruent to either 1(mod3) or −1(mod3).

Hi
I just wanted to reinforce this.

Take 977 then...
977 = 3q' + 1 for some q'
=> 3q' = 976
=> q' = 976/3 = 325.333333333...

but q' is not an integer? How can the above be true?

Reply 1

Hodor

Original post by isint

Hi
I just wanted to reinforce this.

Take 977 then...
977 = 3q' + 1 for some q'
=> 3q' = 976
=> q' = 976/3 = 325.333333333...

but q' is not an integer? How can the above be true?

You've assumed that 977 is congruent to 1 (mod 3). (It isn't.)

(edited 10 years ago)

Reply 2

isint

But it IS congruent to -1.

977 = 3q' - 1 for some q'
=> 3q' = 978
=> q' = 978/3 = 326 = integer

Is this true for ALL primes?

Reply 3

Hodor

Original post by isint

But it IS congruent to -1.

977 = 3q' - 1 for some q'
=> 3q' = 978
=> q' = 978/3 = 326 = integer

Is this true for ALL primes?

Every prime greater than 3 is congruent to either 1 or -1 (mod 3), yes. Try to prove it!

Reply 4

brianeverit

Original post by isint

But it IS congruent to -1.

977 = 3q' - 1 for some q'
=> 3q' = 978
=> q' = 978/3 = 326 = integer

Is this true for ALL primes?

If p is any prime then p-1,p and p+1 are three consecutive integers so one of them must be a multiple of 3 and if p is not equal to 3 then p-1 or p+1 must be divisible by 3 and so must be an integer.