`a-=b` (mod n) and `b-=c` (mod n) implies `a -= c` (mod n)

a-=b (mod n) and b-=a (mod n) are equivalent

Let Z Z be the set of all integers and let m be an arbitrary but fixed positive integer. Show that the relation "congruence modulo m" on Z Z defined by : a -= b '(mod m)' implies (a-b) is divisible by m, for all a,b in Z Z is an equivalence relation on Z Z .

If a≡b (mod n), then state whether a^2≡b^2 (mod n)

44 ≡ 6 (mod n) where n can be

If a equiv b (mod n), then a^3 equiv b^3 (mod n) (write true or false).

If the vectors vec a, vec b, vec c are non-coplanar and l,m,n are distinct real numbers, then [( l vec a + m vec b + n vec c) (l vec b + m vec c + n vec a) (l vec c + m vec a + n vec b )] = 0, implies (A) lm+mn+nl = 0 (B) l+m+n = 0 (C) l^2 + m^2 + n^2 = 0

If x = sum _(n =0) ^(oo) a^(n) , y = sum _(n=0) ^(oo) b ^(n), z= sum _(n=0)^(oo) c^(n) where a,b,c are in A.P. and |a| lt 1, |b| lt 1, |c| lt 1, thenx,y,z are in-

If the sum of n terms of an A.P. is given by S_n=a+b n+c n^2, w h e r ea ,b ,c are independent of n ,t h e n (a) a=0 (b) common difference of A.P. must be 2b (c) common difference of A.P. must be 2c (d) first term of A.P. is b+c

The total number of ways in which 2n persons can be divided into n couples is a. (2n !)/(n ! n !) b. (2n !)/((2!)^3) c. (2n !)/(n !(2!)^n) d. none of these

An operation ** is defined on the set A={1,2,3,4} as follows: a**b,ab(mod5) for all a,binA .Prepare the compositon table for ** on A and from the table show that - (i) multiplication mod(5) is a binary operation, (ii) ** is commutative on A, (iii) is the identity element for multiplication mod(5) on A, and. (iv) every element of A is invertible.