Let n be a fixed positive integer. Define a relation R on the set Z of integers by, a R b iff n | a - b . Then, R is not

Property 7 (DIvision Algorithm) If a whole number a is divided by a non-zero whole number b then there exists whole numbers q and r such that a = bq + r whole either r = 0 or r < b .

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).

Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m , 9m+1 or 9m+8 .

Use Euclids division Lemma to show that the cube of any positive integer is either of the form 9m ,\ 9m+1 or, 9m+8 for some integer m .

Use Euclids division Lemma to show that the cube of any positive integer is either of the form 9m ,\ 9m+1 or, 9m+8 for some integer m .

Show that the cube of a positive integer of the form 6q+r,q is an integer and r=0,1,2,3,4,5 is also of the form 6m+r

The positive integers p ,q ,&r are all primes if p^2-q^2=r , then find all possible values of r

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.