Let x, y ∈ Z. Prove that x ≡ y + 1 (mod 2) if and...

Let x, y ∈ Z. Prove that x ≡ y + 1 (mod 2) if and only if x ≡ y + 1 (mod 4) or x ≡ y + 3 (mod 4)

Solutions

Expert Solution

If you have any doubts ask me in comments ....

I just use basic defenation of mod..and division

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let x, y, z be a primitive Pythagorean triple with y even. Prove that x+y ≡...

Let x, y, z be a primitive Pythagorean triple with y even. Prove that x+y ≡ x−y ≡ ±1 mod 8.

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}....

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}. a) Prove or disprove: A ⊆ X b) Prove or disprove: X ⊆ A c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y ) d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one to one and onto functions. Then g ◦ f : X → Z is one to one and onto; and (g ◦ f)−1 = f−1 ◦ g−1 ).

Mystery(y, z: positive integer) 1 x=0 2 while z > 0 3 if z mod 2...

Mystery(y, z: positive integer) 1 x=0 2 while z > 0 3 if z mod 2 ==1 then 4 x = x + y 5 y = 2y 6 z = floor(z/2) //floor is the rounding down operation 7 return x Simulate this algorithm for y=4 and z=7 and answer the following questions: (3 points) At the end of the first execution of the while loop, x=_____, y=______ and z=_______. (3 points) At the end of the second execution of...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just f (because f is already curried) let f x y z = (x,(y,z)) let f x y z = x (y z)

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.

Let S be the cone z = 4 − ( x^2 + y^2)^(1/2) where z ≥...

Let S be the cone z = 4 − ( x^2 + y^2)^(1/2) where z ≥ 0, oriented with downward pointing unit normal vectors. The image below contains S. The grey plane is the xy-plane. Let F~ (x, y, z) = <− z , x − y , x + y >. Use Stokes’ Theorem to evaluate Z Z S ∇ × F~ · dS~.

If X, Y and Z are three arbitrary vectors, prove these identities: a. (X×Y).Z = X.(Y×Z)...

If X, Y and Z are three arbitrary vectors, prove these identities: a. (X×Y).Z = X.(Y×Z) b. X×(Y×Z) = (X.Z)Y – (X.Y)Z c. X.(Y×Z) = -Y.(X×Z)

let p = 1031, Find the number of solutions to the equation x^2 -2 y^2=1 (mod...

let p = 1031, Find the number of solutions to the equation x^2 -2 y^2=1 (mod p), i.e., the number of elements (x,y), x,y=0,1,...,p-1, which satisfy x^2 - 2 y^2=1 (mod p)

Subjects