What is the number which will be divided by 201 and 671 leaving?
Let us write the question in the language of Modular Airthmetic. Let the number be X. X =6 (mod 201 ) X =8 (mod 671) ________________________________________________ The 1st. congruence can be rewritten in parametric form as X =6 +201 t where t is any integer. Putting against the 2nd congruence, we get 6+201 t =8 (mod 671 ) 208 t =2 (mod 671 )(iii) In Modular Airthmetic, except for some special cases, direct division not allowed as in classical Airthmetic, in stead we take recourse of Modular Inverse and multiply it. 444 is the Modular Inverse of 208 Modulo 671. [ This means , 444*208=1 (mod 671) ] [ Hint :- By Euclidean formulation 671=3 (201)+68 201=2 (68)+65 68 =1 (65)+3 65=3 (21)+21 21=2 (10 )+1 Working back ] Multiplying (iii ) by 444 (the Modular Inverse ), we get t=217 (mod 671 ) t =217 +671 u where u is any integer. X =6+201 t (see above ) X =6+201 (217 +671 u ) X =6+43617+134871 u X =43623 +134871 u Take u =0 X =43623 : One of the Answers. Taking the different values of u ,we can determine infinity numbers as there is no bar as to the range. The Lowest positive ANSWER is 43623. ANSWER.