Determine the length of the period for the decimal expansion of 8881 Note that 8881 83 107. =

Q: Find g.c.f. and l.c.m. of the numbers 2730 and 693. SHOW YOUR WORK! Are the numbers 2730 and 693…

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Q: For which odd primes p is 3 a square mod p?

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Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: prove the formula of Mersenne primes

A: A number of the form is a Mersenne prime if and only if p is prime.To prove this we need to show…

Q: Let p + 7 be an odd prime. Determine the value of the Legendre symbol (7/p).

A: Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a…

Q: How many square roots of 3 (mod 1001) are there? (Hint: 1001 = 7 * 11 * 13)

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Q: If n is a positive integer, explain why each prime in the factorization 'n² must occur an even…

A: (a) Let p be a prime number that occurs in the prime factorization of n2.   This implies that p…

Q: What is the greatest prime you must consider to test whether 5317 is prime? The greatest prime to…

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Q: Prove or disprove: There are inline number of primes

A: Known fact:A prime is a natural number whose factors are 1 and itself.

Q: 259 ompute the Legendre symbole (3), noting that 733 is prime.

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Q: Show that every nonzero residue (mod 17) can be written as a power of 3

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Q: 3 Р 1 ? What if 3 is replaced by = 2. For which primes p is -3, 5, -5, 6, or -6 ? Any patterns? Is…

A: The symbol  is the Legendre symbol, used in number theory to denote whether a number is a quadratic…

Q: Find the prime factorization of 224331. List the product of primes in increasing order (in the form…

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Q: Use the pigeonhole principle to prove that the decimal expansion of a rational number min eventually…

A: To Prove:  Using the pigeonhole principle that the Decimal expansion of a rational number m/n…

Q: Let n (arak-1 ·*a1a0)3 be the base 3 expansion of a positive integer n. Find the base 3 expansion of…

A: Solution: We know that if for any positive integer r have the base 3 expansion (ap,ap-1,.....,a0)3…

Q: Euler's Theorem states that if n is a positive integer and a is relatively prime to n, then ar(n) =…

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Q: c.) There is a conjecture called "Goldbach's weak conjecture" (yes, really!) that every odd number…

A: Given: Odd numbers greater than 5. To prove: Every odd number greater than 5 can be written sum of…

Q: 2. What are the prime factors for the below numbers and arrange them in a canonical form. a. 17 b.…

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Q: The binary expansion 0.abcd... means 23 24 and each binary digit a, b, c, d, ... is either 0 or 1.…

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Q: claim: n? +n+1 is prime for n > 1. either prove the claim is true, or find a counterexample to show…

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Q: Show that every nonzero residue (mod 17) can be written as a power of 3.

A: We show that every non zero residue(mod 17) can be written as power of 3.

Q: 11. Use you the test for divisibility and test for primality to find a pair of twin primes between…

A: We want to find the twin prime between 100 to 106 We know that twin prime are the number differ by…