Question

3. (i) Prove that the set of all linear combinations of a and b are precisely the multiples of g.c.d(a,b). (ii)* Prove that a and b are relatively prime iff every integer can be written as a linear combination of a and b.

Answer 1

The set oy of afb how let с 2 now by 50 p = po (i) dute have to prove that all linear Combination & are precisely the multiples of ged (9,6) p=fal (9,6) it is a fact - 4 a, b z , an niy ez sz. axtby=ged (9,6) proof - lel let 8 = {ax+by/nyez & axtby >0} note that a, bt8 B# $ archemedian property a min. Element of 8 sit. =min(8) how we claim that a p=gel (916) of note that. ano otby for Ho, Yo EZ-0 we claim that pola if of a then - a= p qtr where ovrch 0X2- pq = aquot bago 9-r=pq 9-8= aqrot bago => p= 911-940) +(-9%) b reb & r<min (B) contradiction beet that is an plo Bo pla e plo same as be sit. clad clb det Bince clad cb el contby, fnyez how y el anotbyo = cle gel (a, b) now I here rap AA 介 pla CEZ how AP AT Hip

se linear Combination of afb. a&b are (11) now we have to proce telectively prime iff aury integer can be written as as a prime if 1 a&b are & now inte if now if af b are relatively there are integer xty sit. axtby now let suppose not relatively prime then =) Cab=p>= (by def. of relatively prime) 9 = Ap of b = Bp for some A,B EZ than axtby= A x + Bpy = P(Ax +By) ++ now as we know that p>If theroe doesn't Exist Any ger sit. cp=1 where cez þ> I. so if for some in some integer have integer xey axtby = 1 which is the contradiction. བོད (915) are relatively prime . if (a,b) are relat helatively prime. then a ged (916) = =p=1 uchich proved in part first of this problem. sing (i) proof q niy ez sit axtby=p=1 now we Qo usi
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