What is the remainder when 7^n-6 n is divided by 36 for n=100 ?

Step-by-step solution

Suppose p ( n )=7^ n -6 n =(6+1)^ n -6 n Using Binomial expansion, we get aligned & =1+6 n+ ^n c₂ 6^2+ ^n c₃ 6^3+ . .+6^n-6 n & =1+6^2 ( ^n c₂+ ^n c₂ 6+ .+6^ n-2 ) & =1+36 ( ¹⁰⁰ c₂+ ¹⁰⁰ c₂ 6+ .+6⁹⁸ ) n=100 & =1+36 ~m where m = ¹⁰⁰ c₂+ ¹⁰⁰ c₂ 6+ . .+6⁹⁸ aligned So, when 7^ n -6 n is divided by 36 , the remainder will be 1.