For the first question, have you taken a look at Euclid's method for generating Pythagorean triples?
For the second question, I would let b = 4n...what do you find?
Let A,B,C be integers such that A^2 + B^2 = C^2. Prove at least one of A and B is even.
Please help guide me through this problem so I can understand it better thanks.
I have another question also
let a and b be positive such that a^2 = b^3
given 4 divides b, prove that 8 divides a
Im hoping if someone can guide me through this problem also thanks.
for the first one i want to try to use a proof by contradiction but i am kind of stuck on how to do it
and for the 2nd one when b = 4n
b^3 = (4n)^3
so a^2 = (4n)^3
so a = sqrt((4n)^3)
Hello, gfbrd!
First, we will establish a theorem.
Consider the square of an integer,
If is even, we have: .
. . Hence: .
If is odd, we have: .
. . Hence: .
Therefore, the square of an integer is either:
. . (1) a multiple of 4, or
. . (2) one more than a multiple of 4.
We are given: .
Suppose and are both odd.
We have: .
The we have:
. .
. . . . . . . . .
. . . . . . . . .
Hence: is two more than a multiple of 4.
. . . . . .It cannot equal a square,
Therefore, at least one of and must be even.