Question

Use Euclid's algorithm tof find HCF of 1990 and 1445. Express the HCF in the form 1190m + 1445n.

Answer 1

We know that Euclid's division Lemma is x and y for any two positive integers, there exist unique integers q and r satisfactorily x = yq + r, where 0 ≤ r <y. In case r=0 then y will be the HCF.

1445=1190x1+255
1190=255x4+170
255=170x1+85
170=85x2+0

We have found r=0
Hence, HCF(1190,1445)=85

So, now
85 = 255 - 170
=(1445-1190)-(1190-1020)
=(1445-1190)-(1190-255x4)
=1445-1190-1190+255x4
=1445-2×1190+(1445-1190)x4
=1445-2×1190+1445x4-1190x4
=1445+1445×4-2×1190-1190×4
=1445x5-1190x6
=-1190×6+1445×5
=1190x(-6)+1445x5
=1190m+1445n
(where m=-6 and n=5)