Question

The table of exercise 1 suggests that one of the numbers in any primitive Pythagorean triple is divisible by 4, one is divisible by 3 and one is divisible by 5. Please Prove this

Answer #1

Please keep in mind that the numbers in all primitive Pythagorean triple are not necessarily that one is divisible by 4, one is divisible by 3 and one is divisible by 5.

Here is an example that shows the argument is not correct for all primitive Pythagorean triple

(5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41)

These all are primitive Pythagorean triple but see in first on two numbers are divisible by 4 and 5 respectively but the last one is not divisible by 3.

Similarly in a second primitive Pythagorean triple two numbers are divisible by 4 and 3 respectively but the last one is not divisible by 5. And so on

So the above argument is not valid for the general cases.

Ask if you have any quarries. Thank you

Prove: If (a,b,c) is a primitive Pythagorean triple, then either
a or b is divisible by 3.

Find all Pythagorean triples (not necessarily primitive) in
which 25 is one of the numbers and then prove that the list is
complete.

In class we proved that if (x, y, z) is a primitive Pythagorean
triple, then (switching x and y if necessary) it must be that (x,
y, z) = (m2 − n 2 , 2mn, m2 + n 2 ) for some positive integers m
and n satisfying m > n, gcd(m, n) = 1, and either m or n is
even. In this question you will prove that the converse is true: if
m and n are integers satisfying...

Prove: Let (a,b,c) be a primitive pythagorean triple. then we
have the following
1. gcd(c-b, c+b) =1
2. c-b and c+b are squares

(a) Prove that if y = 4k for k ≥ 1, then there exists a
primitive Pythagorean triple (x, y, z) containing y.
(b) Prove that if x = 2k+1 is any odd positive integer greater
than 1, then there exists a primitive Pythagorean triple (x, y, z)
containing x.
(c) Find primitive Pythagorean triples (x, y, z) for each of z =
25, 65, 85. Then show that there is no primitive Pythagorean triple
(x, y, z) with z...

Prove: If (a, a+1, a+2) is a Pythagorean triple, then a= 3

Please answer the following in FULL detail!
Is there a Pythagorean triple consisting of three Fibonacci
numbers? Give an example if there is one, or a proof if there
isn't.

(a) Use modular arithmetic to show that if an integer a is not
divisible by 3, then a 2 ≡ 1 (mod 3).
(b) Use this result to prove that in any Pythagorean triple (x,
y, z), either x or y (or both) must be divisible by 3

Exercise 6.6. Let the inductive set be equal to all natural
numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 +
n.
(b) ∀n, 4n − 1 is divisible by 3.
(c) ∀n, 3n ≥ 1 + 2 n.
(d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.

1. Prove that an integer a is divisible by 5 if and only if a2
is divisible by 5.
2. Deduce that 98765432 is not a perfect square. Hint: You can use
any theorem/proposition or whatever was proved in class.
3. Prove that for all integers n,a,b and c, if n | (a−b) and n |
(b−c) then n | (a−c).
4. Prove that for any two consecutive integers, n and n + 1 we
have that gcd(n,n + 1)...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 5 minutes ago

asked 6 minutes ago

asked 10 minutes ago

asked 17 minutes ago

asked 19 minutes ago

asked 19 minutes ago

asked 23 minutes ago

asked 28 minutes ago

asked 34 minutes ago

asked 38 minutes ago

asked 38 minutes ago