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 = g.c.d.(a,b).

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 = 41 (f ) 121x – 88y = 572

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.