Prove that if x is even and y is even, then the product of x and y is divisible by 4.

Q: determine whether multiplication by A is one-to- one by computing the nullity of A and then applying…

A: Let (x, y) be in nullspace of A.  Then, A(x, y)t = 1-22-4-36xy = 000     => x - 2y = 0…

Q: In real-number multiplication, if uv = uw and u # 0, we can cancel u and conclude that v = w. Does…

A: To explain: either v=w or give counter example.

Q: Let G be a direct product of H and K then Z(G) is a direct product of Z(H) and Z(K).

A: By the set inequality i have solved the problem

Q: Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

A: Disproof by an example.

Q: For f(x), g(x), and Zn[x]in the given question ,, find the greatest common divisor d(x) of f(x) and…

A: Given: f(x)=x3+2x2+2 and g(x)=2x5+2x4+x2+2 in Z3[x]. To determine: the greatest common divisor d(x)…

Q: Prove that [x] ∈ ℤ/n is a zero divisor if and only if there exists an integer y >1 such that y | x…

A: Let x∈ℤn is a zero divisor. There there exists a nonzero element y such that  xy=0 in ℤn. Hence xy…

Q: If B has a column of zeros, then so does BA if this product is defined. True O False 4

A: Given B has a column of zeros , then to find BA has a column of zero or not .

Q: Prove that |ab|= |a | |b| for any numbers a and b.

A:

Q: True or False? If d = gcd(a,b) , then every common divisor c of a and b divides d.

A:

Q: Show that A = has two different factorizations of the form A = L · U where L is lower triangular…

A: Given: A=0001. To show: A has two different factorization of the form A=L·U, where L is lower…

Q: Prove that no element of ℤ/n is both a zero divisor and a unit.

A: To Determine :-  Prove that no element of ℤn is both a zero divisor and a unit.

Q: - Find a bijection between Z and E (the even numbers), prove your answer is correct.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

A: I have proved using the fact that gcd can be expressed as integers linear combination q

Q: Show that [o 2] A 3 0 has no A = L-U factorization where L is lower triangular and U is upper…

A:

Q: Apply any method to prove that if a|b and alc then a|(bc – 3c)

A: We will use direct proof method to prove the given statement

Q: Prove that (A-1)-1 = A. %3|

A: Let A be inverse of matrix A. Then, by definition We know that

Q: Use the Factor Theorem to prove that x + c is a factor ofxn + cn if n Ú 1 is an odd integer.

A: To prove the given statement.

Q: Show that is isomorphic to Z2.

A:

Q: Let xer" and ge I show that =yex is an imer product.

A: I have to show that the given relation satisfies three conditions to be a inner product

Q: Let * be the operation on the cartersian product of real numbers R *R such that (a,x) * (b,y) =…

A: Given that, let * be the operation on the cartesian product of real numbers R*R  such that (a, x) *…

Q: Prove that x4 + 15x3 + 7 is irreducible over Q

A:

Q: Prove that if A ∈ Mn×n(F) has two identical columns, then det(A) = 0.

A: The determinant is the scalar value computed by expanding a row or a column.

Q: Suppose  v,w is an element of V. Explain why there exists a unique x which is an element of V such…

A: Suppose v, w is an element of V. Explain why there exists a unique x which is an element of V such…

Q: .ABCDE is apentagon.Prove that the resultant of →AB,−→AE,−−→BC,−−→DC,−→ED and AC is 3AC

A: Here we have given a pentagon ABCDE. Consider the Figure for the solution. We have to prove that…

Q: Let and 001 1 0 0 2 1 A = 0 If the entries of A and v belong to Z3, what is the third entry of Av? 2…

A: Recall matrix multiplication and modulo arithmetic.

Q: 2. Suppose someone is trying to illustrate Cantor's diagonalization proof that |X # |P(X)|, where X…

A: A bijectIon means a mapping which is both one - one and onto .

Q: Prove that there are integers x and y such that 10x-13y=1

A:

Q: Show that 2Z ∪ 3Z is not a subring of Z.

A: Theorem for Sebring Test: For any ring R, a subset S of R is a subring if and only if: it is closed…

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: Let P3 have the inner product

A: Given, pt=t2 and qt=2t-3. given by evaluation -2,-1,0,1, and 2. p-2=4q-2=-4-3=-7 p-1=1q-1=-2-3=-5…

Q: Prove that every element of Sn (n > 1) can be written as a productof elements of the form (1k).

A:

Q: If Ua = {(x1, x2, x3)" E R³|x1+ 2x2 + 3ax3 = a} is closed under addition, then a = 0. Prove that the…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Prove that if gcd(a, b) = 1 and cla, then gcd(b, c) = 1.

A: We need to prove If gcd(a,b)=1, then gcd(b,c)=1. The proof is given below.

Q: Prove that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1. %3D

A:

Q: Prove the following statement: If x € Z, shares a common nontrivial factor with n, then a does not…

A:

Q: 5. Let a, b, N E Z where N > 0. Prove that [ab] = [[a][b], where [r] denotes the remainder of r…

A:

Q: Prove: [a] is an element of Zn has a multiplicative inverse if and only if gcd(a,n)=

A:

Q: Prove that √2 is not an element of Q(?).

A:

Q: prove that and = nd(n)/2

A:

Q: Let a ∈ R[[x]] with a0 not equal to 0. Prove that a has a multiplicative inverse in R[[x]]. You may…

A: We have given that , a∈ℝx a = a0 + a1x + a2x2 + ......  where each ai ∈ ℝ. Since , 1 ∈ ℝx  is an…

Q: Show that 2Z U 3Z is not a subring of Z.

A: show that 2Z∪ 3Z is not a subrings of Z:   Solution:-             Let 2∈ 2Z∪3Z…

Q: Prove that given a and b, with b> 0, ther exist unique integers q and r satisfying a- qb•r in the…

A: We will prove the given statement.

Q: Let * be the operation on the cartersian product of real numbers R* R such that Then the (а,х)*…

A:

Q: 2. Determine the greatest common divisor of a(x) = x³ – 2 and b(x) = x + 1 in Q[x] and write it as a…

A: Given the polynomials   ax=x3-2 and bx=x+1 to find gcd(a(x), b(x)) and write it as a linear…

Q: Prove that if a is odd and b is odd, then (a + 1) b is even

A: Number system.. properties

Q: prove that if a>4 then (a-2)^2>4

A:

Q: Determine the number a in Z5 so that (x - a) becomes a divisor of f (x) = x ^ 3 + 2x ^ 2 + 3x + 3 in…

A:

Q: Use the Factor Theorem to prove that x + c is a factor of xn + cn if n ≥ 1 is an odd integer.

A: Let f(x)=xn+cnNow take,x+c=0i.e.x=-cNow, considerf(-c)=(-c)n+cn          =0          (Since, n is an…

Q: For all a and b in Z, a - b is a factor of an - bn. (Hint: ak+l - bk+I = ak(a - b ) + (ak - bk)b.)

A: we have to prove that For all a and b in Z, a - b is a factor of an-bn