when 6^83+8^83 is divided by 49 , tell the remainder left where ^ denotes raised to?PLS EXPLAIN (Acco. to me its 0)......

we can write this question as

(7-1)⁸⁹ + (7+1)⁸⁹.  

open this binomially

=  ( ⁸⁹C₀7⁸⁹ - ⁸⁹C₁7⁸⁸ + ⁸⁹C₂7⁸⁷ +... - ⁸⁹C₈₇7² + ⁸⁹C₈₈7¹ - ⁸⁹C₈₉7⁰) + (⁸⁹C₀7⁸⁹ + ⁸⁹C₁7⁸⁸ + ⁸⁹C₂7⁸⁷ + ...- ⁸⁹C₈₇7² + ⁸⁹C₈₈7¹ - ⁸⁹C₈₉7⁰)

=2*( ⁸⁹C₀7⁸⁹ + ⁸⁹C₂7⁸⁷ + ⁸⁹C₄7⁸⁵ +...+ ⁸⁹C₈₆7³+ ⁸⁹C₈₈7¹)

⁸⁹C₀7⁸⁹+⁸⁹C₂7⁸⁷+⁸⁹C₄7⁸⁵+...+⁸⁹C₈₆7³   will all be divisible by 49 because they consist of 7² as one of their factor but not

so  dividing this 2(⁸⁹C₈₈7¹) by 49

we get   2(⁸⁹C₈₈7¹)%49 = 21 ---Remainder