The remainder when the sum S = _ r=1 ¹⁰⁰ (21)^ r is divided by 400 is

Step-by-step solution

We need to find the remainder when S = _ r=1 ¹⁰⁰ (21)^ r is divided by 400 . Using the binomial theorem, we can expand (21)^r as (1 + 20)^r : (1 + 20)^r = 1 + ^ r C₁(20) + ^ r C₂(20)^2 + ^ r C₃(20)^3 + Since we are dividing by 400 , and 20^2 = 400 , all terms from the third term onwards are multiples of 400 and will leave a remainder of 0 . Thus, modulo 400 , we have: (21)^r 1 + 20r 400 Now, substitute this into the sum S : S _ r=1 ¹⁰⁰ (1 + 20r) 400 S _ r=1 ¹⁰⁰ 1 + 20 _ r=1 ¹⁰⁰ r 400 Calculate the two sums separate