• Aucun résultat trouvé

Let (x, y, z) be a primitive Pythagorean triple

N/A
N/A
Info
Télécharger
Protected

Academic year: 2022

Partager "Let (x, y, z) be a primitive Pythagorean triple"

Copied!
1
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)

MIDTERM: SOME PROBLEMS FROM PREVIOUS YEARS

1. Find all integers x, y satisfying the following equation:

98x+ 68y= 2.

2. Solve the system of congruence equation:

(3x≡1 (mod23) 2x≡9 (mod17)

3. What is the remainder of 2124 when divided by 41?

4. What are all the primes p < 14 for which x12 ≡ 1 (mod p) is valid for all x such that(x, p) = 1?

5. Which of the following functions is multiplicative? Why?

(a) σ(n) (the sum of divisors function);

(b) φ(n)2; (c) φ(n) + 1?

6. Let (x, y, z) be a primitive Pythagorean triple. Is it possible that x≡2(mod 8)? Why?

7. Let p, q be distinct prime numbers, let α, β be positive integers and let n = pαqβ. Assume that the productα·β is even. Is it possible that the number of divisorsτ(n)of nis divisible by four: 4|τ(n)? If your answer is ’yes’, provide an example. If your answer is ’no’, provide an explanation.

1

Références

Télécharger maintenant ( PDF - 1 Page - 73.66 KB )

Documents relatifs

introduction Let (X, d, m) be a compact metric measure space

Mondino, Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds, available at http://arxiv.org/abs/1505.02061v1.... Cheeger, A

If X is a smooth variety, let T∗X be the cotangent bundle of X

In the mid-1980s the second author observed that for a connected reductive complex algebraic group G the singular support of any character sheaf on G is contained in a fixed

Let G be a group, H be a subgroup of finite index, and let Z have trivial G-action

Write a detailled proof of the Cartan-Eilenberg double coset for- mula, using the hints given in the lecture..

Lemma 1. Let X be a complete metric space. Let ρ : X → R

Suppose that G is a finitely generated semi- group such that every action by Lipschitz transformations (resp.. First it must be checked that the action is well-defined. The first one

Let V be a r -dimensional real vector space. lattice Λ. Let X = [a1

Need the analog of Bernoulli periodic polynomials: some locally polynomial functions on alcoves that satisfy some equations. I will call them W not to mix them with the

Exercise 1. Let X be a complex manifold.

Denote by H 0 (X, L) the vector space of global sections of L, and suppose that this space has

Exercice 1. Let J ⊂ k[X 0 , . . . , X n ] be a homogeneous ideal. Show that the radical rac(J) is homogeneous.

Construct a second family of lines contained in

Let G be a nite group, and let p be a prime number.

Let us denote by RG perm p the category of all p {permutation RG {modules. The following proposition generalizes to p {permutation modules a result which is well known for