Given p > 2 a prime number, prove that: (p+1 |("") = 0(modp 5.1) for any natural number 2 snsp-1 ) = (-1)" (mod p) 5.2) for any 0snsp- 1,EN = 2 (mod p). 5.3) [fuggontion: lco E 1 ond E ah

Q: Let n be a positive integer. Prove 41+3n mod 9.

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Q: Prove this claim: For every positive integer d and every integer n, the following equation is…

A: Using division algorithm we solve this problem. The steps are given bellow.

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Prove the following statement or disprove by giving a counterexample "for all positive intergers n…

A: We have to prove or disprove the following statement For all positive integer n and for all integer…

Q: Find a number a such that 0 ≤ a < 111 and (10270+ 1)35 = a (mod 111).

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Q: 2-) Find all in tegers x such thet 5x = 12 (mod 7) 10x = 4 (mod 1) 7x E 16 (mod 10d ond 1occ < x <…

A: Given: 5x≡12(mod 7) 10x≡4(mod11) 7x≡16(mod 10) and 1000≤x≤3000

Q: 2. What is the least residue of 610 C 512 10.4512 (mad 1112

A: Euler's theorem is used

Q: Let be P odd an prime s.t- gcd (y² + 4v₁ p) = 1. 2 If (x + y √B ) ² = u ² + 4v (mod pk) (VK 6 N),…

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Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: (()) b) Use part a) to find an integer that satisfies the congruence x² = -1 (mod 17) a) Prove that…

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Q: For each statement, indicate whether it is true or false. Then prove or disprove it. (a) For all SC…

A: (a) FALSE.  Consider S ⊆ℤ, defined by  S = { 2+8k | k ∈ℤ } U { 2+12r | r∈ℤ }.  Then the given…

Q: п(п+1)(п+2) E}=1jG + 1) = а. 3

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Q: 2. If a = 3 (mod 11) and b = 9 (mod 11), compute the values, positive and fully-reduced modulo the…

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Q: 7. Let n > 1. Suppose that a"-1 = for all non-zero remainders a mod n. Show that n is prime. an-1 =…

A: Given that, n≥1 and an−1≡1modn We have to show that n is prime. We know that Euler's generalization…

Q: 4. Let p be an odd prime. (a) For any integer 1 <k <p-2, show that (+1) = (#¹) where k denotes the…

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Q: Consider p> 2 a prime number, justify that: (P,') = (-1)" (mod p) to any 0 sn sp- 1,n EN. 1.1 1.2…

A: In the given problem we use the definition of combination. Then the problem is solved by…

Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: 2221 (mod 13) is equal to O 10 (mod 13) O 1 (mod 13) 7 (mod 13) 6 (mod 13) O None of these

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: e.g. Student ID=61190003 average=(6+1+1+9+0+0+0+3/8) x 0.01 = 20/8 = 2.5 then round the number to an…

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Q: Find the area of each circle, given the radius r or the diameter d. Leave your answer in tems of t.

A: See the detailed solution below.

Q: Prove 1 0, Ø ! n n=1

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Q: 5. The Fibonacci numbers are defined as: fo = 0 , fi = 1 , and fn = fn-1 + fn-2 , for n > 2. Give a…

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Q: 2. Prove that if ar b (mod n), then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x…

A: We first find the gcd of the given numbers and then use that to solve the given congruence. The…

Q: If the temperature at which a certain compound melts is a random variable with mean value 115°C and…

A: We have to solve given Questions.

Q: 1.21 Theorem. Let a natural number n be expressed in base 10 as  n=akak-1...a1a0 (Note that what we…

A: We know that 10 mod 3 = 1. So,

Q: IfOcy (mod n) and e> 0 then a) *= y (mod cn) b) x= y (mod n) ) x=y (mod) d) None of these :Select…

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Q: mod 1537). 2. Frove the following statements: a) For any integer a, the units digit of a? is 0, 1,…

A: We can solve this expression of  numerical analysis

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: If x00 = 1(mod 109) then possible value of x is a) 108 (b) 218 (c) 109 (d) None of these

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Q: Let me Z+ be odd with prime-power factorization m = p₁¹p2²p3³p4, where a1a2 a3 a4 P₁ = P2 = P3 = ±5…

A: As per the question we are given a odd numbers m and its prime power factorization as m =…

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: Compute 21200 mod 27.

A: We see,         21 ≡ (-6)  mod 27     =&gt; 212 ≡ (-6)2  mod 27                 ≡ 36  mod 27…

Q: Suppose gcd(a, n) = 1. Then a(") = 1 (n

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Q: The least residue of 5500 mod(17) is Select one: O none O 7 O 13 O10 O 8

A: The least residue is, 13 The detailed solution is as follows below :

Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: 1. i) Explain why 3 is a primitive root modulo 14 ii) What is log3(11) mod 14?

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