Question

1. Write a proof for all non-zero integers x and y, if there exist integers n and m such that xn + ym = 1, then gcd(x, y) = 1.

2. Write a proof for all non-zero integers x and y, gcd(x, y) =
1 if and only if gcd(x, y^{2}) = 1.

Answer #1

Prove that for all non-zero integers a and b, gcd(a, b) = 1 if
and only if gcd(a, b^2 ) = 1

For all integers x, if (x ^2 + y^2) is not equal to 0 (mod 4),
then x is odd or y is odd. Write the contrapositive of this
statement. Write the contrapositive of this statement. Write the
negation of this statement. c. Prove this statement using a proof
by contraposition or a proof by contradiction?

Given non-zero integers a, b ∈ Z, let X := {ra + sb | r, s ∈ Z
and ra + sb > 0}. Then: GCD(a, b) is the least element in X.

Write a formal proof to prove the following conjecture to be
true or false.
If the statement is true, write a formal proof of it. If the
statement is false, provide a counterexample and a slightly
modified statement that is true and write a formal proof of your
new statement.
Conjecture: There does not exist a pair of integers m and n such
that m^2 - 4n = 2.

N particles can move in plane (x,y).
Write down coordinates and momenta of all particles forming the
phase space and determine number of degrees of freedom
s.
(a) x1, px1,
y1, py1,
x2, px2,
y2, py2,…,
xN, pxN,
yN, pyN ,
s=2N
(b) x1, px1,
x2, px2, …,
xN, pxN ,
s=2N
(c) y1, py1,
y2, py2,…,
yN, pyN ,
s=2N
(d) x1, y1,
x2, y2, …,
xN, yN ,
s=2N
and why you choose

Say that x^2 = y^2 mod n, but x != y mod n and x != −y mod
n.
Show that 1 = gcd(x − y, n) implies that n divides x + y, and
that this is not possible, Show that n is non-trivial

Write a proof for the statement below by proving the
contrapositive.
If x,y have the same parity then 4 | x2 -
y2

Write a C program that asks the user to enter two integers x and
n. Then the program computes
xn (=x * x * x …… (n times)) using for
loop.
using printf scanf

Find all integers n (positive, negative, or zero) so
that (n^2)+1 is divisible by n+1.
ANS: n = -3, -2, 0, 1

Statement: "For all integers n, if n2 is odd then n is odd"
(1) prove the statement using Proof by Contradiction
(2) prove the statement using Proof by Contraposition

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 8 minutes ago

asked 13 minutes ago

asked 29 minutes ago

asked 30 minutes ago

asked 38 minutes ago

asked 42 minutes ago

asked 48 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago