If `n >1` , the value of the positive integer `m` for which `n^m+1` divides `a=1+n+n^2+ddot+n^(63)` is/are `8` b. `16` c. `32` d. `64`

The greatest positive integer which divides n(n+1)(n+2)(n+3) for all ninN is

The greatest positive integer which divides (n+1)(n+2)(n+3)...(n+r) for all ninN is

If n is a positive integer, show that, 9^(n+1)-8n-9 is always divisible by 64.

If n is a positive integer, prove that |I m(z^n)|lt=n|Im(z)||z|^(n-1.)

The number of ordered pairs of positive integers (m,n) satisfying m le 2n le 60 , n le 2m le 60 is

If n and r are two positive integers such that nger,"then "^(n)C_(r-1)+""^(n)C_(r)=

If m , n are positive integers and m+nsqrt(2)=sqrt(41+24sqrt(2)) , then (m+n) is equal to

If A is a nilpotent matrix of index 2, then for any positive integer n ,A(I+A)^n is equal to A^(-1) b. A c. A^n d. I_n

Consider a sequence {a_n}w i t ha_1=2a n da_n=(a n^2-1) /(a_(n-2)) for all ngeq3, terms of the sequence being distinct. Given that a_1a n da_5 are positive integers and a_5lt=162 then the possible value (s)ofa_5 can be a. 162 b. 64 c. 32 d. 2

Let T_r denote the rth term of a G.P. for r=1,2,3, If for some positive integers ma n dn , we have T_m=1//n^2 and T_n=1//m^2 , then find the value of T_(m+n//2.)