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i assume we cannot consider 0
Note that any number must be 7k + r with r = 0,1,...6
@estudier i think all numbers can be expressed that way
(7k + r)^3 = (7k + r)^2 would that help - i'm clutching at straws to be honest!!
So I think it is 7k+1
"(7k + r)^3 = (7k + r)^2" Maybe expand these instead of setting them equal...
1^6=7*0+1 2^6=7*9+1 3^6=7*104+1 4^6=7*585+1 I mean every number can be expressed as 7k + r with r = 0,1,...6
So, I think u mean to say 7k+1
I mean every number can be expressed as 7k + r with r = 0,1,...6 Yes, I said that above....
oh.....THAT WAS A HINT
The question does say 7k or 7k +1
For 7k, (7k)^2 (mod 7)= 0^2 (mod 7)=0, (7k)^3 (mod 7)=0^3 (mod 7)=0 For 7k+1, (7k+1)^2 (mod 7)= 1^2 (mod 7)=1, (7k+1)^3 (mod 7)= 1^3 (mod 7)=1 For 7k+2, (7k+2)^2 (mod 7)= 2^2 (mod 7)=4, (7k+2)^3 (mod 7)= 2^3 (mod 7)=1 ......... For 7k+6, (7k+6)^2 (mod 7)= 6^2 (mod 7)=1, (7k+6)^3 (mod 7)= 6^3 (mod 7)=6 for any 7k+r, if the result is r, then it has the same form.
@NewbieCarrot Considering certain remainders is a good way to go....
Well x=n^6...... Now, If n=0 MOD 7 then , n^6=0 MOD 7 Thus, x can be expressed as 7k.... Now if n=1,2,3,4,5,6 MOD 7 Then n^6=1 MOD 7 ---> BUT I HAVEN'T PROVED THIS PART.
(7k+r)^2 = 49k^2 + 14kr + r^2 = 7(7k^2+2kr) + r^2
(7k+r)^3 = 343k^3 +147k^2r +21kr^2 +r^3 = 7(49k^3 +21k^2r +3kr^2) +r^3
@estudier You got it.
Not done yet....
7(7k^2+2kr) + r^2 7(49k^3 +21k^2r +3kr^2) +r^3 have the same form with 7k+r while r=r^2=r^3 (mod 7)
OK, the reaminders are the same, and.....?
The only possible answer is r=0 or 1. And we are done
n = 7k+r n^2 = 7x + r^2 n^3 = 7y + r^3
r = [1,6]
"The only possible answer is r=0 or 1. And we are done" True, but a little more explanation would be nice...:-)
let,n = 7k+r (r = 1,2,3,4,5,6) n^2 = 7x + r^2 n^3 = 7y + r^3 plugin r = 1,2,3,4,5,6 and the intersection of remainders gives the form of a number thats both square and cube
Yes, that's right, well done.
Just include r= 0 as well....
oh yes... .
Is there this rule: If n=x MOD y then, n^a = x^a MOD y
Is there this rule: If n=x MOD y then, n^a = x^a MOD y Yes (r>=1)
GREAT....... Then I think I proved if n=1,2,3,4,5,6 MOD 7 Then n^6=1 MOD 7
a^(p-1) = 1 mod p (gcd a,p = 1)
Here is my experience, when you see this sentence "is of form 7k or 7k +1" you always should try all 7k+r to get the results. Same as 3k.
a^(p-1) = 1 mod p (p prime, gcd a,p = 1) (OR a^p = a mod p) Fermat's Little Theorem
Indirectly related to the "necklace2 problem....
But how???? That is related to permutation and combination.
n^p-n strings involving 2 or more colours are partitioned into disjoint sets of p strings each set being strings obtained from each other by by a sequence of cycles ie p divides n^p-n which is Fermat's Little Theorem
Special case of your necklace problem.... Loads of beads of n colours, how many different necklaces of p beads can be made, p is prime?
(n^p-n)/2p + [n(p+1/2) -n]/2 + n
Did u mean that every colour has only one bead.
You make a string of p beads and join the ends, so that's n^p strings. Of those, n strings are beads of 1 colour alone (one for each colour) So n^p-n strings having at least 2 colours etc etc....