{1,4,7,10, 13, 16, 19,...}. An element of T is called irre- ducible if it is not 1 and its only factors within T are 1 and itself. 3. Let T = (a) Suppose that a E T and b e T, and c is a positive integer. Prove that if a = bc, then c E T. (b) Demonstrate that every element of T can be factored as a prod- uct of irreducible elements of T. (c) Perform a sieve to find all irreducible elements of T between 1 and 100. (d) Find three examples of elements of T with nonunique factoriza- tions into irreducibles.

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Let p(x) = x³ – 2x + 4 and compute p(A). 1 A = 0 -4 4 NOTE: Write the elements of the matrir…

A:

Q: Let A = {4n^2 + 1|n ∈ Z}. Show that A is a subset of Z but not equal.

A:

Q: x and y. Prove that if alb and alc, then a|(xb+ yc) for any integers

A:

Q: Show that if gcd (a, y) = 1 and a \ xy Then a\ x

A: Definition : Greatest common divisor ( gcd): Let  a and b be are two  integers, not both 0.  A…

Q: 32. Suppose B1, B2 are o-fields of subsets of 2 such that B C B2 and B2 is countably generated. Show…

A:

Q: prove that 2. catX+c

A: We use that- sin2x + cos2x =1  so that 1-cos2x= sin2x

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

A:

Q: 5. Let A = {x E Z|x = 12r - 5 for some integer r} and B = {y € Z|y=6s +1 for some integer s}. Prove…

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Q: The zero-product principle states that if AB = 0, then________.

A: Zero-product principal:

Q: et A be m x n and let  be in R^n. If A has a row of zeros, show that Ax has a zero entry

A: We have to solve given problem:

Q: Prove that if a eZ then the only positive divisor of both a and a+1 is 1.

A:

Q: 26 a ba Q(1E) = }a+bV2 / a, be Q4 2.1) Show (K,t,') eld under add. and rmult, of amotmias 2.2) Show…

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

A: We use the Division Algorithm to prove the given statement.

Q: 2.4.3. Let p(r1, T2) = 16, T1 = 1, 2, 3, 4, and r2 joint pmf of X1 and X2. Show that X1 and X2 are…

A:

Q: In the field GF(p"), show that for every positive divisor d of n, - x has an irreducible factor over…

A:

Q: e of all polynomials in C[0, ed in C[0, 1] with respect to

A: To prove a subset W is a subspace of V, it must satisfy 3 conditions: 1) 0 ∊ W. 2) closed under…

Q: Suppose X = (X1, X2, . .. , Xk) ~ Multinomial(n, p) , with p = (p1, P2, ..., Pk), prove that Cov(X;,…

A: Introduction :-  Given that X is a vector of random variable and it follows Multinomial(n,p) Where p…

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: the set of all elements in Z, that are not zero fac- tors. An element a e Zn is a zero factor if…

A:

Q: The set of polynomials {1, x² + 3x“, 2x² – 7x4} is a linearly independent set in P4.

A: NOTE: Due to time limit answer of 1 is given here, for other please ask in a different question.…

Q: Find the characteristic of Zm, × Zm, × · · × Zmr where m;'s are positive integers.

A: The ring ℤm1×ℤm2×...×ℤmk is a ring with unity 1,1,1,...k times Now, to find characteristic of…

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: Show that if gcd (a, y) = 1 and a \ xy Then a \ x

A: if 1. gcd(a,b)=1then there exist some integers m and n such that am+bn=12. suppose a and b are…

Q: 1. For every element x in A, there is an element y in B such that (x, y)EF

A: Remark: Let A and B be two non-empty sets. A relation R from A into B is subset of A*B. A function f…

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: 1. Let Z defined by the product of two numbers taken separately from two boxes containing 1, 2,3 and…

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…

Q: verify that (AT )T = A

A:

Q: Prove that, IXI<a Rf and If -aL X Z a

A:

Q: Prove that for any integer a, [a]=m = [r]=m where r is the unique remainder when a is divided by m,…

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Q: 1. (a) Prove that if alb and alc. then alb² + 10c + a.

A:

Q: Consider the polynomials P, (1) = 1 + 1² and p2(1) = 1 – 1². Is {p · P2} a linearly independent set…

A:

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: The zero-product principle states that if AB 0, then

A: If AB = 0Then zero-product principle states that this is possible only if - A = 0andB = 0

Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

Q: Let S = {n + m√2 : n, m ∈ Z}  To prove that the sum of any two elements of S is an element of S.…

A: Given: S=n+m√2:n,m∈Z  (a) Let a+b2∈S for all a,b∈Z and let c+d2∈S for all c,d∈Z Add both these…

Q: Show that if gcd (a, y) = 1 and a \xy Then a\ x

A:

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

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Q: Let z = x + yi, z, = 1- 0.5i, and z, + z, = 3-1.5i. Find In(z,) - In(z2) C) In(50) + 2nin A) In(25)…

A:

Q: Theorem 2.8: For positive integers a and b grd co,b) lem ca,b) =ob Coro llory: Icm ca,b) = ob For…

A: Let us consider two pairs of numbers such as 8, 16 and 3,5

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

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Q: Let a ∈ G be such that |a| = ∞ . For m≠n prove that a^m ≠a^n

A:

Q: 6. If X has full rank, so that P = X(X'X)-1X', prove that C(P) = C(X).

A: The rank of a matrix denotes the number of independent rows or the number of independent columns in…

Q: Let z = x + yi, z2 = 1– 0.5i , and z, + z2 = 3 – 1.5i. Find In(z,) – In(z2) %3D A) In(25) ± 2niN D)…

A:

Q: 1 - Which one of the following is an ideal of Z? a) O Z x 2Z b) O Z x {0} c) O Z2 d) 2Z Boş bırak

A:

Q: True or False? If d = gcd(a,b), then there exists integers x, y such that d = ax +by

A: solution.

Q: Prove that for any integer a, [a]=m = [r]=m where r is the unique remainder when a is divided by m,…

A:

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

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.