Math miscellaneousEasy
Question
The remainder left out when 82n - (62)2n + 1 is divided by 9 is
Options
A.0
B.2
C.7
D.8
Solution
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
= (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
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