Question

10^n+72*n-1. chia hết cho 81

Answer 1

Ta có :10\(^n\)+72n-1
= 10\(^n\)-1+72n
= (10-1)[10\(^{\left(n-1\right)}\)+10\(^{\left(n-2\right)}\)+...+10+1]+72n
= 9[10\(^{\left(n-1\right)}\)+10\(^{\left(n-2\right)}\)+...+10+1] - 9n + 81n
= 9[10\(^{\left(n-1\right)}\)+10\(^{\left(n-2\right)}\)+...+10+1-n]+81n
= 9[(10\(^{\left(n-1\right)}\)-1)+(10\(^{\left(n-2\right)}\)-1)+...+(10-1)... + 81n
ta có 10\(^k\) - 1 = (10-1)[10^(k-1)+...+10+1] chia hết cho 9 =>9[(10^(n-1)-1) +(10^(n-2)-1) +... +(10-1) +(1-1)] chia hết cho 81 =>9[(10^(n-1)-1)+(10^(n-2)-1)+...+(10-1)... + 81n chia hết cho 81 =>đpcm.