I really need help. It's about number theory. Would yo be so kind help me?

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2 Answers
Mar 21, 2018

Answer:

Use the fact that if #d|n#, then #d|n^2# and if (#d|n# and #d|m#), then #d|n-m#

Explanation:

Suppose that #p=1# or #p# is prime, and that #p|a+b# and #p|a^2-ab+b^2#

From #p|a+b# , we conclude that #p|a^2+2ab+b^2#

Therefore, #p|3ab#, so #p|3# or #p|a# or #p|b#.

Now show that if #p# divides #a# pr #b#, then #p = 1#

Finish by showing that #9 cancel(|)(a+b,a^2-ab+b^2)#

Mar 22, 2018

Answer:

See below.

Explanation:

Note that

#gcd(a+b,a^2-ab+b^2) = gcd((a+b)^2, a^2-a b + b^2)#

and also

#(a+b)^2 equiv (a²-a b + b^2) mod 3# because

# (a²-a b + b^2)-(a+b)^2 = 3 a b#

hence

#gcd(a+b,a^2-ab+b^2) = {(1),(3):}#