Integrative Biology 200 PRINCIPLES OF PHYLOGENETICS Spring

to the congruence 15x2 + 19x + 6 ≡ 0 (mod 11) Factor the lefthand side of the congruence; show that a solution of the quadratic congruence is a solution of one of the two different linear congruences ] 14 Find the solutions of the congruence 12x2 + 25x ≡ 10 (mod 11)

Exercices - Congruences

le syst eme de num erotation de base 6 s’ ecrivent x3y Exercice 10 D emontrer que N = n(2n+ 1)(7n+ 1) est divisible par 6 quelque soit l’entier n non nul

Exercice 1

2) v eri ant les equations du syst eme (S 1), v eri e les equations du syst eme : (S 2) ˆ 23x 1 4y 1 + 23y 2 (mod 26) 23x 2 19y 1 + 11y 2 (mod 26) (b) A l’aide de la partie B, montrer que tout couple ( x 1; x 2) v eri ant les equations du syst eme (S 2), v eri e les equations du syst eme (S 3) ˆ x 1 16y 1 + y 2 (mod 26) x 2 11y 1 + 5y 2

13 Congruences - NIU

29 Solve the congruence 42x≡ 12 (mod 90) Comment: You need to recall Theorem 1 3 5, which states that ax≡ b(mod n) has a solution if an only if gcd(a,n) is a divisor of b Also note that the congruence is stated modulo 90, and so the most satisfying answer is given in terms of congruence classes modulo 90

GRADE 11 ADVANCE MATHEMATICS 114: GEOMETRY

GR 11 ADVANCE MATHEMATICS M4 CONGRUENCE AND SIMILARITY 7 11 4 1: BASIC GEOMETRIC CONCEPTS Undefined terms are basic terms in geometry which cannot be defined but can be described Some undefined terms are points, lines and planes These undefined terms form the basis of our study of the subject and for defining other geometric terms

Math 4

of the individual equations From exercise 7(b) of Assignment 3, the congruence x2 0 ≡ 1 mod 2 α0 has 1 solution if α0 = 1, 2 solutions if α0 = 2, and 4 solutions if α0 ≥ 3 Noting that the congruence x2 0 ≡ 1 mod 1 has one incongurent solution modulo 1, we deduce x2 0 ≡ 1 mod 2 α0 has 2e solutions with e as deﬁned above If p is

Quadratic Congruences - Xavier University

Recall Exercise 3 4 10(a) There we showed (by completing the square) that solving the general quadratic congruence € ax2+bx+c≡0(modp) for an odd prime p (with (a,p) = 1) is equivalent to solving the simpler congruence € y2≡Δ(modp), where Δ € =b2−4ac (the discriminant of the quadratic) and x and y are related by the linear

Section 37 - Linear Diophantine Equations

Exercise 6: For which integers c with 0 ≤ c