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: 1. If p>9 2 5 and p and q are both primes, prove that 24 p - q.

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Q: A. If p + 2 is a prime number, prove that p? + 3 is divisible by 4.

A: Let p≠2 be a prime number.

Q: Show by contraposition that for all integers n^2, if  is odd, then n + 1 is even.

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Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

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Q: ) Give the contrapositive of the following statement concerning a positive integer p: If p > 2 and p…

A: Contrapositive of a conditional statement of form  (p^q)→r is  ~r→(~pv~q)

Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

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Q: Prove that if n is a positive integer, then 133 divides 11"+1 + 122n-1

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Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

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Q: Suppose that the prime factorizations of two positive integers x and y are expressed using a common…

A: The prime factorization is breaking down a number into the product of only prime numbers. These…

Q: Prove that Vp+1-\p is irrational where p is prime integer.

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Q: For which prime p,  p^2+2 is also prime?

A: Put p=3 in p2+2 and obtain that, 32+2=9+2=11. Clearly, 11 is a prime number. Thus, p=3 satisfies…

Q: 24. Show that if p is a prime and k is an integer such that 1sksp-1. then p divides ().

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

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Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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Q: If p is a prime number greater than 3, which of the following represents the sum of the positive…

A: we will check all the options to ensure which is correct.

Q: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

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Q: 11. If p is a prime and p divides 4.":,, thenp divides " for some i.

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Q: Show that if p > 1 and p divides (p − 1)! + 1, then p is prime.

A: Proof by contradiction: Assume p is not prime ⇒p is composite

Q: Show that there is a prime number p such that p+4 and p+6 are also prime numbers.

A: Prime number is a number p having only two divisors p and 1 only.

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

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Q: Suppose that p is a prime. Prove that p | α if and only if p | α2.

A: Since, If p|α , this is obvious that p|α2.

Q: If a and b are relatively prime then alb

A: Given below the detailed solution

Q: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

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Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

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Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: А. If p 2q 5 is a prime number, prove that p2-q? is divisible by 3.

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Q: Prove that a positive integer n is p

A: Topic :- Number system

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 +q2 + r2.

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Q: Prove that if p> 3 is a prime number, then 24 | (p2 - 1).

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Q: 2. If p > q > 5 and P and q are both primes, prove that 24 p - q² |

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Q: If p2q25 is a prime number, prove that p-q² is divisible by 8.

A: Given, p≥q≥5 are prime numbers, Now, p2-q2=p+qp-q. Since, p and q are prime which are grater than…

Q: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: Given: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even

Q: Prove that if p is a prime integer, then 3√p is irrational.

A: To Prove that: If p is a prime integer, then 3√p is irrational.

Q: Show that the sum of twin primes p and p + 2 is divisible by 12, provided that p> 3.

A: Let p, p + 2 be twin primes. Then p *(p+1) + 1 = p^2 + 2p + 1 = (p+1)^2 ,  a perfect square.

Q: Let p and q be prime numbers. (1) If p is not equal to q, how many generators does Zpq have? (2) If…

A: Given p and q be primes.

Q: Suppose p and q are distinct prime numbers. Find the divide x. 1. number of positive integers r < pq…

A: Given that p and q are distinct prime numbers. To find: number positive integer x ≤pq such that…

Q: if p> q > 5 and p, q are both primes, prove that 24 | p² – q

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Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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Q: Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

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Q: Use induction to prove: for any integer n > 1,4 evenly divides 9"

A: for any integer n≥1, 4 evenly divides 9n-1

Q: Show that if a and b are relatively prime then (a+2b,2a+b)=1 or 3.

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Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 + q2 + r2.

A: We have to prove that 3 divides p2+q2+r2, where p,q and r primes other than 3.

Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

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Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

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