The remainder left out when 82n- (62)2n+1 − is divided by 9 is

Question

The remainder left out when 82n- (62)2n+1 − is divided by 9 is (A) 0 (B) 2 (C) 7 (D) 8

Tardigrade Admin · Accepted Answer

82n-(62)2n+1=(1+63)n-(63-1)2n+1 =(1+63)n+(1-63)2n+1=(1+nc163+nc2(63)2+....+(63)n)+(1-(2n+1)c163+(2n+1)c2(63)2+....+(-1)(63)(2n+1)) =2+63(nc1+nc2(63)+....+(63)n-1-(2n+1)c1+(2n+1)c2(63)+....-(63)(2n)) ∴ Reminder is 2

Q. The remainder left out when $8^{2 n} - (62)^{2 n + 1}$ − is divided by $9$ is

$0$

$2$

$7$

$8$

Q. The remainder left out when 82n−(62)2n+1 − is divided by 9 is

0

2

7

8

82n−(62)2n+1=(1+63)n−(63−1)2n+1 =(1+63)n+(1−63)2n+1=(1+nc1​63+nc2​(63)2+....+(63)n)+(1−(2n+1)c1​63+(2n+1)c2​(63)2+....+(−1)(63)(2n+1)) =2+63(nc1​+nc2​(63)+....+(63)n−1−(2n+1)c1​+(2n+1)c2​(63)+....−(63)(2n)) ∴ Reminder is 2

Q. The remainder left out when $8^{2 n} - (62)^{2 n + 1}$ − is divided by $9$ is

$0$

$2$

$7$

$8$