1. (i) Show that 7 is a primitive root modulo 41. (ii) Find all solutions mod 41 of the congruence x12 = 23 mod 41.

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

Q: if b is modulo m and a is another unit with a=b mod m, we say that d is the---------of a modulo m to…

A: As with such a large amount of concepts we are going to see, congruence is straightforward, perhaps…

Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

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Q: The equation 82r = 12 (mod 13) has O 1 incongruent solution modulo 12 O 4 incongruent solutions…

A:

Q: Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has…

A: Given p is a prime number and 6≡0 mod p. Therefore either p=2 or p=3. Let d=gcda,p. We know that if…

Q: 6. (i) Solve the congruence relation 3x + 5 = 2(3 – 5x) (mod 35) for r. - (ii) If a = b (mod n) and…

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Q: Find the smallest positive integer x that solves the congruence: 7x≡5 (mod52)

A: 7x≡5 (mod52)---(1)

Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: (7) (Z/50Z)*. Which of the other elements of (Z/50Z)* are also primitve roots modulo 50? Find a…

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Q: 1. How many solutions( mod 419) does x = 186 mod 419 have? (Show your work!) 2. How many solutions (…

A: Given: x2≡186 mod 101 We need to solve x2≡186 mod 101 and how many solution exists.

Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p => αp-1…

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: If m = 2* is a power of 2, explain how you could use repeated squaring to compute am (mod n) for all…

A: a) We have m = 2k. First we illustrate the method of repeated squaring to compute am (mod n). Then…

Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: a) How can an inverse of a modulo m be used to solve the congruence ax ≡ b (mod m) when gcd(a,m) =…

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Q: 1. Find all simultaneous solutions of the congruences: (a) X= 3 mod 5 x= 4 mod 6

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Q: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm,  10,00,000= 16*62,500 and since 17 is prime and 17 is not…

Q: 3. Use Wilson's Theorem to prove 6(k - 4)! = 1 (mod k:), if k is prime.

A: Wilson's Theorem: Let k be a prime number, then by Wilson's theorem, (k-1)!=-1(modk)

Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: 1. Show that an integer n is divisible by 2 if and only if its last digit is even. (Hint: For each i…

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Q: Find all solutions mod 31 to the congruence z = -5 (mod 31).

A: We will use python code to find the solutions Now x5 ≡  -5 (mod 31)  ≡  26  (mod 31) And we only…

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this…

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Q: 4. (a) Solve the set of congruences х 1(mod 3), x= 2(mod 4), x = 3(mod 5).

A: We  will solve it by Chinese Remainder Theorem Given  x≡1(mod3)x≡2(mod4)x≡3(mod5) Compare it with…

Q: It is given that 3023 is prime, and 3023 = 2(1511) +1 where 1511 is also prime. (a) How many…

A:

Q: show that 2 is a primitive root mod 3^e for all e>=1

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Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n>1

Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: I know that using a corollary that the congruence x2 ≡ 0 (mod p) that the only solutions are + and -…

A: As per the question we have the following congruence relation : n ≡ 2(mod 4) And we have to prove…

Q: 6), then the congruence has a unique solution modulo p. 11. Show that the congruence x³ = 3 (mod 19)…

A: 11. To prove that the congruence x3≡3 mod 19 has no solutions, whereas x2≡11(mod 19) has three…

Q: Use your work from problem 2 and the method used in the example at the end of Lecture 16 to solve…

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Q: The equation 12x=24 (mod 48) has: * no solution modulo 48 24 non-congruent solutions modulo 48 None…

A: Given the equation 12x≡24(mod 48)

Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

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Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

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Q: (c) Find all integers of order 6 mod 13 and all primitive roots mod 13. Show your working in both…

A: This is a question of Number Theory.

Q: Use the index system modulo 19 to find all solutions of x³ = 11 (mod 19) and 5ª = 4 (mod 19)

A: In the theory of congruence, Fermat's little theorem is very significant. It says that for any prime…

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A:

Q: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

Q: In the congruence equation 3x + 12 = mod 10, which of the following is FALSE O a. 5, 15, 25, 35,..…

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Q: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

A:

Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: 10. Find the sum of all incongruent solutions of a =1 (mod 17). (Enter the least non-negative…

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Q: Find all solutions mod 31 to the congruence x° = -5 (mod 31).

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Q: The equation 21x = 49 (mod 77) has: O no solution modulo 77 only 11 non-congruent solutions modulo…

A: Here we have given equation,  21x = 49 (mod 77)    Let a, b and m be integers such that m>0 and…

Q: The equation 21x = 49 (mod 77) has: O no solution modulo 77 only 7 non-congruent solutions modulo 77…

A: Solution

Q: Find the smallest positive integer xx that solves the congruence: 11x≡5(mod57)

A:

Q: What is the smallest positive integer x in the equarion: 1=3x+2(mod 7)?

A: We have to solve the equation 1 = 3x + 2 (mod 7)⇒1 - 2 - 3x = 0 (mod 7)⇒-1 - 3x = 0 (mod 7)⇒3x = - 1…

Q: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False

A: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.