Find a Pythagorean triple such that A) c^2=3k+1, a^2=3L+1, b^2=3m+1 B) c^2=3k, a

Question

Find a Pythagorean triple such that A) c^2=3k+1, a^2=3L+1, b^2=3m+1 B) c^2=3k, a^2=3L+1, b^2=3m C) c^2=3k, a^2=3L+1, b^2=3m+1
Using the division algorithm that says a^2 for any integer a is of the form 3k or 3k+1. If no such triple exists, give a brief explanation why. Find a Pythagorean triple such that A) c^2=3k+1, a^2=3L+1, b^2=3m+1 B) c^2=3k, a^2=3L+1, b^2=3m C) c^2=3k, a^2=3L+1, b^2=3m+1
Using the division algorithm that says a^2 for any integer a is of the form 3k or 3k+1. If no such triple exists, give a brief explanation why. A) c^2=3k+1, a^2=3L+1, b^2=3m+1 B) c^2=3k, a^2=3L+1, b^2=3m B) c^2=3k, a^2=3L+1, b^2=3m C) c^2=3k, a^2=3L+1, b^2=3m+1
Using the division algorithm that says a^2 for any integer a is of the form 3k or 3k+1. If no such triple exists, give a brief explanation why.

Explanation / Answer

(a)

NOT possible

a^2 = 3L+1, b^2 = 3m+1

=> c^2 = a^2 + b^2 = 3(L+m) + 2

so c^2 cant be in 3k+1 form

(b)

NOT possible

a^2 = 3L+1, b^2 = 3m

=> c^2 = a^2 + b^2 = 3(L+m) + 1

so c^2 cant be in 3k form

(c)

NOT possible

a^2 = 3L+1, b^2 = 3m+1

=> c^2 = a^2 + b^2 = 3(L+m) + 2

so it cant be in 3k form

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