Show that if a and b are relatively prime positive integers, these there exist integers m and n such that `a^(m)+b^(n) -=1` (mod ab).

Show that the square of any positive integer is of the form 3m or, 3m+1 for some integer m .

Statement 1: Let A ,B be two square matrices of the same order such that A B=B A ,A^m=O ,n dB^n=O for some positive integers m ,n , then there exists a positive integer r such that (A+B)^r=Odot Statement 2: If A B=B At h e n(A+B)^r can be expanded as binomial expansion.

Given that sum_(k=1)^35 sin5k^@ = tan(m/n)^@ , where m and n are relatively prime positive integers that satisfy (m/n)^@<90^@ , then m + n is equal to

Show that 7^(n) +5 is divisible by 6, where n is a positive integer

Show that cube of any positive integer is of the form 4m, 4m+1 or 4m+3, for some integer m.

If n is positive integer show that 9^n+7 is divisible 8

Show that n^2-1 is divisible by 8, if n is an odd positive integer.

If n is an even positive integer, then a^(n)+b^(n) is divisible by

Show that the square of any positive integer cannot be of the form 6m+2 or 6m+5 for some integer q.

If a ,\ m ,\ n are positive integers, then {root(m)root (n)a}^(m n) is equal to (a) a^(m n) (b) a (c) a^(m/n) (d) 1