if-a-b-c-Z-and-a-2-b-2-c-2-then-3-ab-

Question Number 219289 by Nicholas666 last updated on 22/Apr/25

i f a, b, c \in Z,

a n d a^{2} + b^{2} = c^{2},

t h e n 3 ∣ (a b) = ?

Commented by Nicholas666 last updated on 22/Apr/25

t h a n k s

Answered by A5T last updated on 22/Apr/25

Any perfect square is equivalent to 0 or 1 (\mod 3)

\Rightarrow a^{2} + b^{2} \equiv 0 + 0, 0 + 1, 1 + 0, 1 + 1 (\mod 3)

But a^{2} + b^{2} is also a perfect square, hence cannot

be equivalent to 1 + 1 = 2 (\mod 3) .

\Rightarrow (a, b) \equiv (0, 0); (0, 1); (1, 0) (\mod 3)

\Rightarrow At least one of a, b is divisible by 3 .

Hence, 3 ∣ (ab)

Answered by vnm last updated on 22/Apr/25

l e t a = k p, b = k q, w h e r e p, q a r e c o p r i m e

o n e o f t h e n u m b e r s p, q c a n b e r e p r e s e n t e d

a s m^{2} - n^{2} a n d t h e o t h e r a s 2 m n

i f 3 ∣ m o r 3 ∣ n t h e n 3 ∣ (2 m n) \Rightarrow 3 ∣ (a b)

i f 3 ∤ m a n d 3 ∤ n

t h e n m = 3 r + s, n = 3 t + u

w h e r e ∣ s ∣=∣ u ∣= 1 \Rightarrow 3 ∣ (m^{2} - n^{2}) \Rightarrow 3 ∣ (a b)

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