Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder is either 1 or prime.

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: Let m be the least common multiple of a and b. Prove that if p is a prime divisor of m, then either…

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Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

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Q: 16. All positive integers with negative squares are prime.

A: It is a statement. And it is  vacuously true.

Q: Prove by induction that it is possible to pay, without requiring change, any whole number of…

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Q: W found that 2,3,5,7 and 11 are not factors of 211. Can we conclude that 211 is prime? Why or why…

A: To check whether 211 is prime number on not.

Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: Show that if m is a positive integer, then p(m²) = m· ¢(m). Hint: consider the prime factorization…

A: Introduction: The phi function of Euler denoted by ϕ is one of the most important number-theoretic…

Q: Let p and q be primes, Prove that plq if and only if p = = 9.

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Q: Let n be an integer. Prove that if n is not divisible by 3, then n² has remainder 1 when divided by…

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Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

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Q: 10. Find all primes p such that 29p+1 is a square.

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Q: Prove by contradiction that there is no largest composite number.

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Q: Let p be a prime number with p>=5. Prove that p must either be one more or one less than a multiple…

A: We use a fact that any number greater that 5 must be written as one of the form…

Q: If A = {x : x is a natural number },               B = {x : x is an even natural number}  C = {x : x…

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Q: Construct a sieve of Eratosthenes for the numbers from 2 to 100. (Write all the integers from 2 to…

A: The given numbers,(a) (b) (c) (d) The objective is to determine whether the numbers are prime or…

Q: .Prove that if n is an odd number then 3n + 1 is an even number. Use direct proof. 2.Prove that…

A: Given n is odd number