Prove that for all integers a, b, and c, with a ̸= 0, if a|b and a|c, then a|(bx + cy)

Q: Let a be an integer such that a = 4 mod 12. Use the definition of mod to prove that a² = 4 mod 12-

A: Solution-:                           a≡4 mod 12a2=(4)2 mod 12     =16 mod  12     =4   mod   12

Q: Prove or disprove each of the following

A: You have to be careful. The big-O is kind of like ≤≤, not like ==, but people forget that. For the…

Q: Prove or disprove that if a|bc, where a, b, and c are positive integers and a ≠ 0, then a|b or a|c.

A:

Q: Give a proof by cases that x ≤ |x| for all real numbers x.

A:

Q: Prove by Contradiction: For all integers x and y, x^2 - 4y is not equal to 2

A: Answer :

Q: Prove that A → B E (Vx)A → (Vy)B, if y is not free in B.

A:

Q: 1. Let F = {AB → C, B → D, CD → E, CE → GH, G → A}. Using Armstrong's axioms, give proofs for the…

A: Here in this question we have given some FDs and using armstrong axiom we have to proof some FDs

Q: Show that u is a zero of F(x) if and only if u is a fixed point of f (x) f(x) = x + a F(x). Suppose…

A: For the first part of the problem, A function  is given. Now, We can say  as a fixed point of  if…

Q: 10, and 1. Suppose that a1 = 5, a2 ak = ak-1 + ak-2 for k = 3, 4, . . Use strong induction to prove…

A: Please upvote. I am providing you the correct answer below.

Q: If ax2 +2hxy+by2 =0 represents two straight lines such that the slope of one line is twice of the…

A: Given in the question the eqaution of line is ax2 + 2hxy + by2  =  0, Slope of one straight line  =…

Q: Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following…

A:

Q: 1. The relationship between the average temperature on the earth's surface in odd years between 1981…

A: code for linear interpolation is given below. code is written using scipy…

Q: Prove the following equalities using equivalence laws: a. ¬(¬rA s) ^ (r V s) =r b. r→(-s→t) =…

A: Let's understand step by step :   Proving : Answer (a) : Given : ¬(¬r∧s) ∧  (r∨s) ≡ r   Now…

Q: b) What happens inside PC-1(Permuted choice 1) and PC-2(Permuted choice 2)? How they are different…

A: ANSWER:-

Q: If x(n) is an absolutely summable sequence with a rational z-transform that has poles at z = 1/2 and…

A: x(n) can be a right sided signal as x(n) is an absolutely summable sequence ROC  must include full…

Q: Let's look at the elliptic curve Y=X + 7 mod 23. As a result, this is the group related E23 (0,7).…

A: Reply

Q: To prove that the Chandy-Lamport algorithm passes the safety condition, demonstrate that it does.

A: Given: Each distributed system has a number of processes running on a number of different physical…

Q: Simplify the following proposition: (P=Q)^(P=¬Q)

A: Given Data :  (P -> Q) ^ (P -> ¬ Q)

Q: A = {[(¬P → Q) AR] V ¬Q} → (QAR)

A: Answer: Answer is given in step 2

Q: f6 = a'b+ ab' = a+ b fg = a' b' ||

A: Proved the validity of the given set of functions, reduced to minimum SOP form

Q: Prove that ¥xay p(x, y) # 3yVx p(x,y)

A: Given that, ∀x∃y p(x,y)≢∃y∀x p(x,y) It contains two variables x, y and by using these two variables…

Q: 1. Prove that for any k>0, nk e o(2v).

A: Solution:-.       1.).    Any k>0,n^k€2^√n

Q: By using the Big-O definition and formal proof, show and prove that a function y=n*+3 can't belong…

A: For a given function g(n), we have set O(g(n)) = { f(n) : there exist positive constants c and n0…

Q: Let A- (1,2 ,3}. Give P(A)

A: We have a given set A and we have to find out the power set of A i.e., P(A). Power set: Power set is…

Q: Prove or disprove:  x↑(y↑z)=(x↑y)↑z, for all x,y,z ∈Z

A: The question is to Prove or disprove:  x↑(y↑z)=(x↑y)↑z, for all x,y,z ∈Z

Q: Suppose that a1 = 5, a2 =10, and ak “ ak-1 + ak-2 for k =3,4,.... Use strong induction to prove that…

A: strong induction to prove that 5 is a factor of an for all n ≥ 3

Q: AES uses operations performed over the finite field GF(2^8) with the irreducible polynomial x8 + x4…

A: Effective polynomial representation: In this case is a GF(2), it will be exclusive OR (XOR) and…

Q: Prove for all P Q: prop (~Q -> ~P) - > (P -> Q) using Coq programming language. additional axioms…

A: COQ'S Reasons: We've seen a lot of different instances of factual statements (assertions) and how to…

Q: Use the Pumping Lemma with length to prove that the following language is non-regular: L = {a²b" a",…

A: Given language is, L={a2bnan, n>0} Pumping Lemma of regular language states that, if the infinite…

Q: If ax2+2hxy+by2=0 represents two straight lines such that the slope of one line is twice of the…

A: If ax2+2hxy+by2=0 represents two straight lines such that the slope of one line is twice of the…

Q: For each of the following  Prove them  formally and identify constants c and n0 if not then disprove…

A: a) Given function is, f(n)= n2+9n+32 The function contains variable n whose degree is 2 and constant…

Q: Find ( axb) x c if a =i-2j+k, b=i+2j-4k, c=2i-3j +k Q2.

A: Answer : I attached an explanation please have a look once.

Q: Assume that A is reducible to B. Show that if B is solvable in polynomial time, then A is also…

A: Answer is given below-

Q: 4. given y → u p → w prove [(p v q) ar] → (x V ¬y) using natural deduction

A: Answer : prove [(pv q) ^r] → (x v¬y) using natural deduction

Q: Prove that m2 = n² if and only if m = n or m=-n. 7) Show that p → q and¬q → ¬p are logically…

A: The product of two negative numbers is always positive. If we find the square of a negative number,…

Q: By contradiction, prove that ∀ x, y ∈ Z, if XY is even and y is odd then x is even.

A:

Q: Show your work.) Suppose the irreducible polynomial of GF (24) is P(x)=x4+x+ 1. Find x3* (x3+x²+1)…

A: ANS 21: OPTION - D

Q: Find the value of x for the following sets of congruence using the Chinese remainder theorem. a. x ≡…

A: Answer

Q: Suppose that a bank machine can dispense money in either 3$ or 10$ bills. Show that any amount over…

A: Clearly the amounts $3, $6, $9, $10, $12, $13, $15, $16, $18 can all be made using 3-dollar bills…

Q: Show that x^5y^3+x^4y^4+x^3y^5 is Ω(x^3y^3).

A: since above question has 2 variables consider big-omega notation for two variables Big omega…

Q: Prove or disprove each of the following: [10] 1. (n+a) = 0(n°), for any real constants, where b > 0.…

A: Here in this question we have given theta notation and omega notation.ans we have asked to prove or…

Q: Show that 2n  E  O(3n).

A: Show that 2n  E  O(3n).

Q: Let A = {(N1, N2) | N1 and N2 are NFAS and L(N1)N L(N2) = Ø}. Show that A is %3D decidable.

A: Given that, A={ <N1, N2> | N1 and N2 are NFAs and L(N1)∩L(N2)=∅} That means N1 and N2 are…

Q: or each of the following subsets of {0, 1}∗ tell whether or not it is regular. Prove your answer.…

A: Given subset is  {0, 1}∗ a) L=  {x | #1(x) = 2.#0(x)}  In the above language there is comparison of…

Q: Prove that (xỹ + Xy) = Xỹ + xy.

A: I have solved it handwritten

Q: Prove by CASES: Let x be an integer. If 3 does not divide x, then 3 divides x2−1.

A: In this question, we have to prove that if a number x is not divided by 3 then x^2-1 will be divided…

Q: the determinant and Solve for Value of w, x, y, z Given: w – 12x +13y – 7z = 23 2w + 15x + 16y +…

A: Given: w – 12x +13y – 7z = 232w + 15x + 16y + 12z = 19 4w – 10x + y + 17z = 24 3w + 2x – 3y + 2z =…

Q: Show using induction that for every x € {a, b}* such that x begins with a and ends with b, x…

A: As per our guidelines we are supposed to answer  only one question. Kindly repost other questions as…

Problems on numbers

Number theory is one of the most appealing topics in mathematics. Given one or more numeric values, a numeric algorithm performs some computation. The use of number theory in computing is important for applications like in the case of cryptographic algorithms. Number theory is also used in various applications such as calculating hash functions or in information security. Euclid’s algorithm is one of the oldest number-theoretic algorithms which is used to calculate the greatest common divisor of two numbers.