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

Q: In how many ways can we select 5 people out of ?10, of which a particular person is not included 9C5…

A: given data  select 5 people out of 10 available people.  no. of ways to select if particular person…

Q: stion 12 Suppose that In 2 = a and In 5 b. Use prop

A: Suppose ln2=a and ln5=b, Given expression is We will use the following properties of log to express…

Q: Let p ≥ 2 be a prime. Suppose p does not divide a. Then ap−1 ≡ 1 (mod p).

A: The given problem is to show the given following modular congreunce that let p is a prime and p…

Q: Given that a =3 and 1 if a is even; a n+1 5а - 1 if a_is odd. Then tern number four, i.e., a а, is

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Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: If p is an odd prime, then (p – 2k)! (2k – 1)! = ..mod p (-1)k+1 (-1)k This option This option None…

A: We prove the following statement with the help of Wilson's theorem

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: (10) The following element is prime in (Ze,+,.) = .... .. (а) 2 (b) 4 (c) 12 (d) No Choice (а) (b)…

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Q: Let p E Z be a prime number and set Zp = {- e Q : If gcd(n, m) = 1, then p m}. a. Show that Zp 1 Q.

A: Solution

Q: mca's a Ik lasb) zd then 巧 is a) Poime c) re latively prime d) of these. none

A: We are given that a,b=d We need to find ad,bd Now, a,b=d means gcda,b=d Now, gcda,b can be written…

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: The elements of order 4 modulo 29 are O 6 and 12 O 7, 14, and 21 O 12 and 17 O None of the choices

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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 =…

A: We have to solve given problem:

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: Which of the following is not correct ? o gcd(e,(p+1) (g –- 1)) = 1 de = 1 (mod (p – 1) (q- 1))…

A: From the given statements

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: 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 c be a nonzero real number and cn? an Which one of the following 5n2 – 3n + 7 statements is…

A: Fourth option is correct.

Q: In how many ways can we select 5 people out of ?10, of which a particular person is not included 965…

A: Given, there are 10 people and 5 are selected.but a particular person is not included.

Q: Disprove the following statements. (a) If n and m are natural numbers, then nm is composite. (b) For…

A: I have given an answer in step 2.

Q: 6. Let pe Z be prime and n E Zt. Prove that the largest power of p which divides n! is given by…

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Q: If n is prime, then 2" – 1 is prime.

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Q: The order of the integer 8 modulo 13 is: O 6 O 4 O None of the choices O 2

A: To find The order of the integer 8 modulo 13.

Q: in is 1+4n eDi where n is an integer. equal to the above 1+471 e2Di where n is an integer. equal to…

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Q: The elements of order 4 modulo 29 are O 6 and 12 O 7, 14 and 21 O 12 and 17 None of the choices

A: To find elements of order 4 modulo 29, we shall check each of these given options one by one. We…

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: 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: Question 7 * If n is a product of twin primes, then a) о(m)o(n) 3 (п - 1) (n — 3) b) Ø(n)o(n) с)…

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Q: Given that P(n) is the equation 1+3+5+7+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we…

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Q: Question 3. Let n > 2 be an integer. Show that n is either a prime or it has а prime faclor < Vп.

A: Question: Let n be an integer such that n≥2. Prove that either the number is a prime number or it…

Q: describe (a,p)

A: Given, if p is a prime, and a is a positive integer.

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: (10) The following element is prime in (Ze, +,.) = %3D ..... (a) 2 (b) 4 (c) 12 (d) No Choice (a)…

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Q: Show that if n is prime and b is a positive integer with n ̸∣ b, then n passes Miller’s test to the…

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Q: Theorem 11.3. If p and q = 2p+1: primes, then either q | M, or q| M,+2, but are not both.

A: As per Bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Question 5: Prove that if nEN is such that 2" - 1 is a prime number, then n must be a prime number…

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Q: 15/If n and n +2 are a pair of twin primes, establish that o (n +2) = 0 (n) +2; this also holds for…

A: σ (n) = sum of all the positive divisors of n

Q: If p is a prime number such that p divides a for some a e Z, then p divides a. Select ane: True…

A: Given that If p is a prime number such that p divides a^2 for some integer then p divides a.

Q: Let c be a nonzero real number and cn? an Which one of the following 5n2 - 3n + 7 statements is…

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

A: A tetrahedral number corresponds to a 3-Dfigure in the shape ofa tetrahedron made of spherical…

Q: Given that P(n) is the equation 1+3+5+7++ (2n − 1) = n², where n is an integer such that n ≥ 1, we…

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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: if p> q > 5 and p, q are both primes, prove that 24 | p² – q

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Q: Consider the statement P given by: "For all n E Z, if n mod 4 1, then n2 mod 4 = 1." (a) Is P true…

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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: Let c be a nonzero real number and an Which one of the following 13n2 – 3n + 7 statements is…

A: here option (e) is true because

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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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.