Two different forms of solutions of Linear Diophantine Equation ax + by = c
If (xo,yo) is a solution of the Linear Diophantine equation ax + by = c , then the set of solutions of the equation consists of all integer pairs (x,y), where either x = xo + tb/d and y = yo – ta/d ( t = ……..,-2,-1,0,1,2,……..) or , x = xo – tb/d and y = yo + ta/d ( t = ……...,-2,-1,0,1,2,…….) where d = ...continues
The Linear Diophantine Equation
Find the general solution ( if solution exist) of each of the following linear Diophantine equations: (a) 2x + 3y = 4 (d) 23x + 29y = 25 (b) 17x + 19y = 23 (e) 10x – 8y = 42 (c) 15x + 51y ...continues
A proof and a solution involving a Diophantine equation
Show that the Diophantine equation x^2-y^2=n is solvable in integers if and only if n is odd or n is divisible by 4. When this equation is solvable, find all integer solutions.
Finding a representative of a congruence class
Find a representative of the congruence class [1143]^-1 in Z mod 1957. ([1143]^-1 is the inverse of some other congruence class).
Theory of Numbers - Euclid's Division Lemma
Theory of Numbers - Euclid's Division Lemma (a) Prove that if a and b are odd integers , then a2 - b2 is divisible by 8. (b) Prove that if a is an odd integer, then { a2 + (a + 2)2 + (a + 4)2 + 1} is divisible by 12. See attached file for full problem description.
Number Theory - Polynomial Congruences Find all solutions of each of the following congruences . (a) x2 + x +1 ≡ 0 (mod 11) ...continues
Chinese Remainder Theorem : Problem of Sun-Tsu
Find all solutions of the problem of Sun-Tsu. Find all integers x such that the remainder after division by 3 is equal to 2, the remainder after division by 5 is equal to 3, the remainder after division by 7 is equal to 2.
Chinese Remainder Theorem and Proofs
The Chinese Remainder Theorem (CRT) applies when the moduli ni in the system of equations x≡ a1 (mod n1) … x≡ ar (mod nr) are pairwise relatively prime. When they are not, solutions x may or may not exist. However, the related homogeneous system (2’), in which all ai=0, always has a solution, namely the trivial s ...continues
Congruences : Multiplicative Inverse
In Z135, the element [4] is a unit. Why? Find the multiplicative inverse [4]^-1 (Please see attachment for proper format.)
Proof of sum convergence for relative primes
See attached file
