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

Q: a) Find C and k that show 8x? + 5× is O(x?). Show your algebra reasons. b) Use proof by…

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Q: Take any arbitrary integer n and then square it, which one is true? n2 cannot be in the form 7k + 5.…

A: Solution is given as follows :

Q: If n has a prime factorization n = pi*p2² ... p" then the sum Eajn d³P(d) is equal to: a1. az

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

Q: 6. Let p, q be two primes. Write n := p.q. Suppose a is coprime to n. Prove that a(m)/2 = 1 mod n.

A: As per our guidelines we supposed to solve the very first question when multiple questions are…

Q: There are infinitely many primes of the form 4k + 3, where k E N. False O This option True This…

A: The given problem is to check whether the given statement is True or False. Given statement There…

Q: 16.K. If c > 0 and n is a natural number, there exists a unique positive num- ber b such that b" : E…

A: Solution for 16K: If n=1, we can take y=x. So let n>1. Let A=t∈ℝ:t>0 and tn<x Claim: A≠∅…

Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

A: We have to solve given problem:

Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

A: twin prime numbers are the prime numbers which are separated by difference of 1 for example 11 and…

Q: Let p be a prime number. For any keN, by Lemma 3.2.9 we have that o(p") = p* -p%-1. (a) Let a be a…

A: The result ϕpk=pk-pk-1 holds true for any natural number k and p being a prime.  The notation a|b is…

Q: (a) Give an inductive proof that the Fibonacci numbers Fm and Fm+1 are reatively prime for all m 20.…

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Q: If p and q .are odd primes, then a) -4 is a primitive root of q. b) 4 is a primitive root of q. c)…

A: Answer : a) -4 is a primitive root of q.

Q: a) If d is an integer such that d | n + 5 and d | n² + 1, then d | 26. b) If n is a multiple of 26,…

A: a. Given that d is an integer such that d|n+5 and d|n2+1.

Q: Which of the following is true? If p is prime and ab| p then a = 1 or b = 1 If lcm(a, b) = b then…

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Q: 5. Suppose p is a prime. Let E := Q[Sp, V/2] where (p = e2™i/P E C. Prove that [E : Q] = p(p – 1).

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Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

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.

Q: If n > 2 is prime then n | a"-1 1 for all positive integers a with gcd(a,n) = 1. Select one: O True…

A: True

Q: Tpand q are win primes, then which of the following is trae Lp+qh divisible by 3 Pq =n-1 for some…

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Q: Let p be a prime number. Prove that (p -1)! a power p + (3p power2  +1)a is divisible by p, for any…

A: Consider the provided question, (a)

Q: Which of the following correctly define what a prime number is? An integer p > 1 such that if d is…

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Q: Which of the following is cyclic? Justify. a. The set of positive integers under usual addition b.…

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Q: 3 Derive the following formula for the sum of triangular numbers, attributed to the Hindu…

A: We have triangular number tn=n+12 ⇒tn=n(n+1)2 We have to prove t1+t2+t3+...+tn=n(n+1)(n+2)6

Q: The number of primes of the form n² – 6n + 5 where n is an integer is O 2 O 3 O 1

A: We have n2 -6n+5 ⇒n2-5n-n+5⇒n(n-5)-(n-5)⇒(n-1)(n-5)

Q: In 10-14, find how many solutions there are to the given equa- tion that satisfy the given…

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Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

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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: If p is a prime number, then which of the following is true. Select one: (p – 3)! = 1(mod p) О (р-…

A: Let p be a prime number. We have to choose the correct option from the given list. Consider the…

Q: Which of the following is cyclic? Justify. a. The set of positive integers under usual addition b.…

A: For (e), {6Z, +}, where 6Z is the set of multiples of 6. Notice that &lt;1&gt; can generate the…

Q: Show that for any odd prime p, the Gauss, G(p), can be rewritten as p-1 G(p) = 5². x=0

A: To Show- Show that for any odd prime p, the Gauss, Gp, can be rewritten as Gp = ∑x=0p-1ζpx2.

Q: 2. ) Show that the cube of any integer is of the form 7k or 7k +1 b) Verify that if an integer is…

A: 2) a. To show that ''The cube of any integer is of the form 7k or 7k±1''

Q: a. For all m, n integers show that (m3 п) тod (m — п) %3D т? + mn + n? b. Prove or disprove: Vn is…

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

Q: Prove: Let p be a prime number and m EN such that m <p². Then if p = 3 (mod 4) then the equation m =…

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Q: Wh of the following are twin primes. 41 & 643 b) 497 & 499 c) 673 & 675 d) None of these

A: Twin prime numbers: Two prime numbers are called twin primes if there is only one composite number…

Q: a. If p E P is a prime number, prove that the cube root of 2p is irrational.

A: We will use the basic knowledge of number theory and logic to answer this question. Note that only…

Q: Which choice gives an example that supports the conjecture, and a counterexample that shows the…

A: The objective is to select the correct option from the given options. Given that, for any real…

Q: Evaluate |P(A)| if A={n: n is a prime number from 1 to 100}

A: To evaluate : PA if A=n : n is a prime number from 1 to 100

Q: If |G = 30 and |Z(G)| that |G| 5, what is the structure of G/Z(G)? What is the structure of G/Z(G)…

A: Solution:-

Q: (2) Let p be an odd prime and suppose that m e Z is not divisible by p. (1) Show that E() \p- = 0.…

A: (1) Given that p is an odd prime. The aim is to show that ∑a=1p-1ap=0. Now take, ∑a=1p-1ap,…

Q: Use trial-and-error to determine the smallest integer n for which the given statement is true. 6n <…

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

Q: tn be a positive integer and M = [2 01 %3D en M" 2n equal to the above None of the mentioned

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Q: Which of the following is true? There are infinitely many primes of the form n2- 9. If lcm(a, b) = b…

A: there are given options. there are infinitely many prime of the form n2-9 it is not true as…

Q: Wh of the following are twin primes. 41 & 643 b) 497 & 499 c) 673 & 675 d) None of these :Select one…

A: Option (a) is correct

Q: Let p be a prime number that does not divide an integer a. Show that: gcd (a, a+p )=1and Icm (a,…

A: Given that , P is a prime number and Pxa where a is an integer .Suppose that ,…

Q: The cube of any integer may has the form: 9k + 6 O This option All of the choices O This option 9k…

A: As per Euclid's division lemma, for two integers a and b there exist two unique integers q and r…

Q: 6) If p is an odd prime, then 1P + 2P + 3P + .. + (p – 1)P = ??(mod p) a) 0 b) 1 c) - 1 d) None of…

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Q: If p is a prime with p> 2, then 2p + 2 is a multiple of O a. 6 b. 4 Oc. None O d. 3 or 4 O e. 2 ar 5

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Q: 5. Let p be a prime. Show that for all integer a we have that pla" + (p – 1)! a and p|a + (p – 1)!…

A: Note: We are required to solve only the first question, unless specified   Theorem Used: Wilson's…

Q: 5) If p is an odd prime, then 2(p – 3)! = ?? (mod p) a) 2 b) 1 c) - 1 d) - 2

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Q: Let N be an integer, and let p be a prime > 7. Suppose p| (N2+7). Show that p is of the form 1+ 7k,…

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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.