How do you fiNd all positive integers m for which the following congruence is true. 100 ≡ 50(mod m)

Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: (a). Find all possible integers k such that the following linear congruence has a solution. (9k? +3k…

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Q: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: ​Q3// Use Chinese theorem to solve the following congruence: X ≡ 2 mod 3 X ≡ 1 mod 5 X ≡ 2 mod 4

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Q: Let m be fixed positive integer. Then an integers a is congruent to an integer b modulom, if: ab|m…

A: Given that m is a fixed positive integer. a and b are integers. Also, given that integer a is…

Q: Solve the following congruences. Make sure that the number you enter is ≥0 and ≤N−1, where N is the…

A: According to the given information, it is required to solve the following congruences. For part (i)

Q: Find all integer solutions for each congruence: Show all your work. a) y-2 ≡ 3 mod 5   b) 2x - ≡…

A: For the second problem 2 has no inverse under the mod 8 operation so not solvable

Q: 4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod…

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

Q: True or False 18 ≡ 12 (mod 3) is not a true congruence.

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Q: *discrete math* Prove that for all integers a, b, if a and b are congruent to 1, mod 3, then their…

A: An integer n is congruent to 1, mod 3, if and only if there is an integer k such that n = 3k + 1.

Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: 3. (а) Using the Legendre symbol, determine your own prime number Pi and non-zero coefficients…

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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: 12. i)Prove by mathematical induction that: 9"- 1 is divisible by 8, for all positive integers n.…

A: according to our guidelines we can answer only three subparts, or question and rest can be reposted.…

Q: Establish that if a is an odd integer, then 2" aʻ = 1 (mod 2" *2) for any n 2 1.

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Q: Use the QRT to explain what ' 35 mod 4' means. There exists a unique integer r such that 3514 There…

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Q: ac bc (mod mc). 14. Show that if a, b, and c are integers with c>0 such that a = b(mod c), then (a,…

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Q: brvib - If m is composite, what is the value (m – 1)! mod m in the standard residue system? Conclude…

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Q: Use mathematical induction to prove that 3n+ 7n−2 is divisible by 8 for every non-negative integer…

A: The given statement is, Pn=3n+7n-2 Let us assume that the statement is true for n=1 P1=31+71-2=8 As…

Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: Use Induction to prove that 22n+1 + 32n+1 is divisible by 5 for all n > 0.

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Q: (a) Let p E N be a prime number, p # 5, and let nE N. Determine the number of solutions to the…

A: First note that this question is only solvable for certain primes and not for all primes. We…

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: Let a and b be integers and n a positive integer. Assume also that a and n have a common divisor d…

A: Use the following concepts, to prove the required result. If a divides b then b is a multiple of a.…

Q: If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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Q: (b) Let x be an integer such that 0 < x < 8 and (10500 + 105000 ) = x (mod 9). Justifying your…

A: We have to find the required value of x .

Q: Let 1Sasb be two positive integers. How can I prove or disprove that there exists a 1sks b-1 such…

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Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

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Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

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Q: Prove or disprove. There is a prime of the form p = 6k – 1 such that the congruence 13x? + 7x +1 = 0…

A: We have the given Quadratic function is f(x)=13x2+7x+1 The Discriminant is given by…

Q: Prove both subparts  (i ) Prove that if p is a prime, then (p − 1)! ≡ −1 (mod p). (ii ) Show that…

A: If P is prime then to show that (p − 1)! ≡ −1 (mod p). And 2nd part (ii) we prove using the first…

Q: Prove Exercise 4.6.12 in the course notes. Then, assuming that gcd(a, n)|b, determine how many…

A: We have to prove that if gcd(a,n) | b, then ax≡b (modn) has no solution. We will prove this by the…

Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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Q: Suppose a and b are relatively prime positive integers. Is it true that (a+2b)4(ab) = 1 (mod ab) ?…

A: We try to solve the question using Euler's Theorem. The detailed solution typeset in LaTeX is…

Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: (5n)! 40 n! is natural number for every natural number n. Prove that

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Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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Q: Let p be an odd prime number. Prove the following statements. (a) (7) = 1 if and only if p = 1, 3…

A: ANSWER

Q: Using Fermat's Little Theorem, solve the congruence 2. x425 + 4 · x108 – 3 . x² + x – 4 = 0 mod 107.…

A: Consider the provided question, 2·x425+4·x108-3·x2+x-4≡0  mod 107                  ......1 Using…

Q: e ab then th

A: Given: ak^2+bz^2 =c mod p To prove the given.

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

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Q: 5. Show that (a + b)n = a + bn mod 2 for all integers a, b, and n with n > 0. Hint: Try looking at…

A: Given data: (a-b)n= an +bn mod 2 for all intergers a,b and n with n&gt;0.