Question

The remainder left out when 82n - (62)2n+ 1 is divided by 9 is here 2n is power of 8 and 2n+1 is power of 62​

Answer 1

      The reminder left out is 2.

• As per the question, 8^(2n)=(8²)^n=64^n=(1+63)^n=63k+1 for k∈N
• So, (62)^(2n+1)=(63–1)^(2n+1)=63m-1 ,m∈N
• So, 8^(2n)-62^(2n+1) =(63k+1)-(63m-1)
• So , 8^(2n)-62^(2n+1)  ={63k-63m}+2
• So, 8^(2n)-62^(2n+1) =63(k-m)+2
• So, 8^(2n)-62^(2n+1) =9×7(k-m)+2
• So, 8^(2n)-62^(2n+1) =9p+2 ,p∈N.
• ∴ the left out remainder is 2.