Show that for any two integers a, b , (a+b)^2 ≡ a^2 + b^2 (mod 2)

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In summary, the given equation (a+b)^2 ≡ a^2 + b^2 (mod 2) is proven by showing that 2 divides 2ab, which can be done by finding an integer k such that 2k = 2ab. This proof generalizes to any prime number.
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Simkate
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Show that for any two integers a, b , (a+b)^2 ≡ a^2 + b^2 (mod 2)

I have my solution below i wanted someone to help chekc if i have done anything wrong. Thank You for your help.

The thing that is going on here is that 2x = 0 (mod 2) for any x. If x = ab, then 2ab = 0 (mod 2).
We see that (a+b)^2 = a^2 + 2ab + b^2. Then this equals a^2 + 0 + b^2 (mod 2).

Therfore, the Proof:

By definition, showing (a+b)^2 = a^2 + b^2 (mod 2) is equivalent to showing that

2 divides [(a+b)^2 - (a^2+b^2)]
2 divides [(a^2 + 2ab + b^2 - a^2 - b^2]
2 divides 2ab

To show 2 divides 2ab we need to find an integer k such that 2k = 2ab. Take k = ab. Thus it is proved.
 
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This looks like a good proof.


However... you already know modular arithmetic, right?
We see that (a+b)^2 = a^2 + 2ab + b^2. Then this equals a^2 + 0 + b^2 (mod 2).
Then this is a complete proof.



(For the record, you could have also proven it by exhaustion -- there are only four cases, and they're easy to check by hand. However, the argument you used generalizes to replacing 2 by any prime)
 

FAQ: Show that for any two integers a, b , (a+b)^2 ≡ a^2 + b^2 (mod 2)

What does the notation "≡" mean in the equation (a+b)^2 ≡ a^2 + b^2 (mod 2)?

The notation "≡" means "is congruent to" or "is equivalent to." In this context, it means that the left side of the equation is congruent to the right side with respect to the modulus 2.

Can you provide an example to illustrate this equation?

Sure, for example, let's take a = 3 and b = 7. Then, (3+7)^2 = 100 and 3^2 + 7^2 = 58. When we take the modulus 2 of both sides, we get 0 ≡ 0, showing that the equation holds true.

What is the significance of using modulus 2 in this equation?

Modulus 2 means that we are dividing by 2 and looking at the remainder. In this context, it allows us to determine whether a number is even or odd. Since 2 is a prime number, using it as a modulus in this equation helps us to identify patterns and relationships between even and odd numbers.

Is this equation only applicable to integers?

Yes, this equation is specifically for integers. In order for the equation to hold true, a and b must be integers.

How is this equation useful in mathematics or other fields?

This equation is useful in various fields, such as number theory, algebra, and computer science. It helps to identify patterns and relationships between even and odd numbers, which can be applied in problem-solving and algorithm development. It also has applications in cryptography and error correction codes.

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