IIT JEE , AIEEE Forum - Mathematics , Algebra

ananth_patri

106^{85 _}85¹⁰⁶divideed by 7...wats the remainder....Plz some one solve it and reply fast to get rates...........

sandeepramesh

0. use modulos and one step :)

ananth_patri

sory i didnt get u.....

sandeepramesh

dont worry if u didnt but the ans is 0 :)

Also search for a similar question in this forum, sboosy has given an alternative soln for a similar sum :)

ramkumar_november

here is one general result ........

aⁿ - bⁿ is divisible by a - b ..........(1)

let x = 106⁸⁵ - 85¹⁰⁶

x = 106⁸⁵ - 85¹⁰⁶ + 1 - 1

x = (106⁸⁵ - 1⁸⁵ ) - ( 85¹⁰⁶ - 1¹⁰⁶)

106⁸⁵ - 1⁸⁵ is divisible by 106 - 1 i.e 105 which is divisible by 7 ( from(1))

85¹⁰⁶ - 1¹⁰⁶is divisible by 85 - 1 i.e 84 which is divisible by 7 (from(1))

so x is divisible by 7

so 106⁸⁵ - 85¹⁰⁶ is divisible by 7

remainder is 0 .

computer001

106

1 mod 7
so (106)^85

(1)^85 mod7=1mod7
85

1 mod 7
(85)^106

(1)^106 mod7=1mod7
so (106)^85=7k+1
(85)^106=7j+1
so (106)^85-(85)^106=7(k-j)
hence reminder=0