Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

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Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

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Step 1

It has been given to us that a, b, c are integers and a2 + b2 = c2.                                                     And we need to prove that for the above equation to be true at least one of a, b is to be even.                                                                                                                                                                                       We can prove this by contradiction as below: -                                                                                           Let us consider that both of a and b are odd integer.

Then we can write a = 2k + 1 and b = 2p + 1 for integers k and p.

Therefore, we will have the calculation as below: -

Step 2

Now we will consider two different cases:

Case I: If c is even: then c is divisible by 2 and so c2 is divisible by 4.

However, a2 +...

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