Question

Please show how to do a proof for this problem. Thank you!

Let mı, m2 be positive integers with gcd(mı, m2) = d. Prove that the system of congruences x = r1(mod mı), x = r2(mod m2) has a solution if and only if rı = r2(mod d).

Answer 1

Answer: - Given that + let mi, my be positive integers with gcd (m, ma)= d. * To prove that the system of congruences - 2=8, (modmi), a = 82 (modmy) has a folution if and only if riso2 (mod d): - → proof : - * Now, god ( m m2) = 0 re, ami+bm2 =d; a,bez > * The congruences are rar, (modm,) and 4282 (modma) 1e, X= Kimitri > X=K3 m2 +82 + * Now, subtract and ine, 2-2= kim, tr, -K₂ m2-82 O= Kim, K2 m₂ +81-82 ③ * compare equations & © azki b=-K2 d= 82-81 * From that dtr-820 (or) : am, tbn2=d -> km Kimd7

real 0 - Kd try-12 Where Kal or other integere also posible. * From 9 d+81-82=0 81= - dtrz (01) 8,=-kdtrz Therefore, This equation is equivalent to 8=82(modd) 8E82 (modd) ] Hence, it is proved.