Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Suppose $a$ , $b$ and $c$ are integers. If $a^2 + b^2 = c^2$ (i.e., $a^2 + b^2$ is a perfect square), show that $a$ or $b$ is even.

View step-by-step help

View step-by-step help

Complex conjugate theorem

Full solution

Q. Suppose

a

,

b

and

c

are integers. If

a^2 + b^2 = c^2

(i.e.,

a^2 + b^2

is a perfect square), show that

a

or

b

is even.

Assume odd integers: Assume both $a$ and $b$ are odd integers.
Express $a$ and $b$ : Let $a = 2k+1$ and $b = 2m+1$ , where $k$ and $m$ are integers.
Calculate $a^2$ : Calculate $a^2$ : $(2k+1)^2 = 4k^2 + 4k + 1$ .
Calculate $b^2$ : Calculate $b^2$ : $(2m+1)^2 = 4m^2 + 4m + 1$ .
Add $a^2$ and $b^2$ : Add $a^2$ and $b^2$ : $4k^2 + 4k + 1 + 4m^2 + 4m + 1$ .
Simplify the sum: Simplify the sum: $4k^2 + 4k + 4m^2 + 4m + 2$ .
Factor out common factor: Factor out the common factor of $4$ : $4(k^2 + k + m^2 + m) + 2$ .
Identify contradiction: Notice that the sum is $2$ more than a multiple of $4$ , which cannot be a perfect square since all perfect squares are either $0$ , $1$ , or $4$ modulo $4$ .
Conclusion: Since our assumption leads to a contradiction, at least one of $a$ or $b$ must be even.

More problems from Complex conjugate theorem

Question

Find the degree of this polynomial.

\newline

`5b^{10} + 3b^3`

\newline

Posted 2 months ago

Question

\newline

(9a + 5) - (2a + 4)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

-3v^2(v^2 - 9)

\newline

Posted 1 month ago

Question

Find the product. Simplify your answer.

\newline

(v - 3)(4v + 1)

\newline

Posted 2 months ago

Question

Find the square. Simplify your answer.

\newline

(3y + 2)^2

\newline

Posted 2 months ago

Question

Divide. If there is a remainder, include it as a simplified fraction.

\newline

(24t^2 + 36t) \div 6t

\newline

Posted 2 months ago

Question

Is the function

q(x) = x^6 - 9

even, odd, or neither?

\newline

\newline

\text{[[even][odd][neither]]}

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

-3q^2(-3q^2 + q)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

(r + 3)(4r + 2)

\newline

Posted 2 months ago

Question

Find the roots of the factored polynomial.

\newline

(x + 7)(x + 4)

\newline

Write your answer as a list of values separated by commas.

\newline

x =

Posted 2 months ago