Let a = IT-1P" where the p, 's are distinct primes. Then A) pila for each i. B) If bla, then b II-1 P with 0 s mi Sn for each i. C) If alc, then pilc for each i. %3D D) All of the above.

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A: The objective of the question is determine the irrational numbers in the given sets.

Q: Abstract Algebra. Answer in detail please.

A: To prove the statements about the given field extensions, (two quadratic and one 4th degree over Q)

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A: Euler's phi function is widely used in number theory and algebra. This function represents the…

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A: To prove the given statements by using mathematical induction.

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A: True

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A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

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Q: (b) Show that įn is real, even though i or 9i is not.

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Q: Q prove that for ony natural number n,2+4 +2n=1/n+1) %3D

A: The solution is given below.

Q: The ideal generated by (1,0) is prime in Z×Z. Select one: O True O False

A:

Q: The value of a prime p such that 19p +1 is a square is a 17 Ob 23 Oc None Od 19 Oe e 17 or 23

A: A detailed solution is given below.

Q: Let p be a prime number. Find all solutions of the equation xP = x in Zp.

A: We have, xp = x  For x = 0 the above equation will be always true. Hence x = 0 is the answer.

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A: Given that p is odd prime. This implies that p is not equal to 2.

Q: 8) wnich of the fohowing is an equa tion ina+ will find the O valu e of 2 in tDe figure on the right…

A: in given right angle triangle

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A: We solve this problem by rearranging the triangles in octagon.

Q: (a) Characterize the primes p for which 63 is a quadratic residue.

A: Quadratic residue: Let p be any odd prime member and gcd(a, p)=1. Then a is said to be a quadratic…

Q: A shop gives gift cards in nominal 25 and 40k. determine the possible total amount that can be…

A: Proof at first, the amounts seem semi-random:

Q: The Catalon numbers defined 2n n=0,12.. ntl in form tor nzi pcove that (n con be given inductivele…

A: Given: Catalan number is defined as Cn=1n+12nn=2n!n!n+1!, n=0,1,2..... Now, we will Prove…

Q: Let p and q be distinct primes with p<q and q# 1mod(p). Show that if G|= pq then G Z/pqZ.

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Q: Show that if a, = (a" – B"), where a = (1 + v5)/2 and ß = (1 – V5)/2 , then th An = an-1 + an-2 and…

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Q: Show that if n = p1p2⋯pk, where p1, p2,…, pk are distinct primes that satisfy pj − 1 | n − 1 for j =…

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Q: z)It Hl show that for ony pasitive integer n, 2.1 ony atar tor t mt or

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Q: 3. If a, beFand ab, show that x + a andx+ bare relatively prime in FIx. 4 (a) Let fly) drle ETyl If…

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Q: how many pairs of primes p and q are there such that q-p=3? Prove your answer.

A: Here, we need to find all the possible pairs of prime numbers p and q such that the difference (q-p)…

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A: NOTE :  As per Bartleby instruction when more then one question is given only one has to be…

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

Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

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Q: 5 = t(n + 1) = t(n +2) = t(n + 3) holds for n = 3655 and 4503. (a) Verify that t (n) (b) When n =…

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Q: A primitive Pythagorean triple is an ordered tripe of integers (x, y, z) such that x? + y² = z²,…

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Q: prove that it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

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Q: Use direct proof to show that a. if m and n are odd integers their sum m+n is even. b. If n is odd…

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Q: Find integers x and y such that gcd(N, 11353) = Nx + 11353y.  Let N = 116023

A: The greatest common divisor(gcd) of a prime number and any other number is always equal to 1.

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Q: the pth term of AP is a and the qth term is p,prove that it's nth term is (p+q-n)

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Q: Show that the n-th tetrahedral number O_n is given by S_n= n(2n^2+1)/3

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Q: 2. Take the last four digits ABCD of your student identification number. If this number is odd let d…

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Q: T(n) is an odd integer if and only if n is a perfect square. a(n)i on oda intese

A: Since you have asked multiple questions, we will solve the Question Q7 (a) for you. If youwant any…

Q: Solve for z. In 63 = In z + In 7 Enter your answer in the box. z =

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.