Suppose a is a positive integer and p is a prime. Prove that pla if and only if the prime factorization of a contains p.

Q: Prove that pqr is a deficient number, where p, q, r are three distinct primes with p <q < r

A: We have to prove that pqr is deficient number where p, q and r dustinct primes  Such that p &lt;  q…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: Prove that for any integer n, n6k - 1 is divisible by 7 if gcd(n, 7) = 1 and k is a positive…

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Q: Prove that if n is an odd integer then 7n+5 is even.

A: Topic:- algebra  Note :- if more than one question uploaded than only first question need to be…

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: Use the Chinese remainder theorem to show that an integer a, with 0 ≤ a &lt; m = m1m2⋯mn, where the…

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Q: Let p be a prime number and a be a positive integer. Prove that p | (a^p - (p-2)!a).

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Q: If p &gt; 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p &gt; 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

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: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: Show that if p is an odd prime, then n = 2º – 1 passes Miller's test base 2 -

A: Given that n=2p-1 where, pis odd prime. Since, 2p is an even integer. Then, n=2p-1 is an odd…

Q: Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? + 1 is divisible by 10.

A: We Shall solve one question only un per the company guidelines as you have asked more than one Que.…

Q: Number theory.

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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: Let a, b be integers for which a01 and ba02 are relatively prime. Show that a and b are also…

A: Given: a and b are two integers for which a101 and b202 are relatively prime. We have to show that a…

Q: Suppose p is prime and a is a non-zero element of Z,. Prove that xP – x + a has no zero in Z,

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Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: Prove that if a > 1, then there is a prime p so that a < p< a! + 1.

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Q: Prove that if p is a prime, then the square root of p is irrational.

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Q: Prove that if n is an integer and 5n + 6 is even, then n is even.

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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: Prove: that if a is an odd integer, then 4 is a factor of a^2 -1

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Q: Prove that the fourth power of any integer has one of the forms 5k or5k + 1, where k is an integer.

A: We need to prove that any integer has one of the forms 5k or 5k + 1, where k is an integer.Let p be…

Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

Q: m and n are relatively prime if and only if am+bn=1 for some integers a and b.

A: It is given that :am+bn=1--(1) for a,b,m,n ∈ Z Now, if  k divides both m and n, then there exist p…

Q: Prove that if m and n are integers and mnis even, thenm is even or n is even.

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

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

Q: Prove that for any prime positive integers P,VP is an irrational number

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Q: Suppose that m and n are relatively prime and r is any integer. Showthat there are integers x and y…

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Q: Recall that a is a type 1 integer if there exists an integer k such that r = 3k + 1. Prove that if a…

A: x is a type 1 integer. So, x = 3k+1, for some integer k. So, x² = (3k+1)²

Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

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Q: Prove that a positive integer n is p

A: Topic :- Number system

Q: Show that if p is an odd prime, then −1 is a quadratic residue of p if p ≡ 1 (mod 4), and −1 is not…

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Q: Suppose that [E:Q] = 2. Show that there is an integer d such thatE = Q(√d) where d is not divisible…

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Q: Prove that if a > 0, k > 1, and ak -1 is prime, then a = 2 and k is prime. %3D

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Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A: Fermat's Theorem : If p is any prime and a is any integer then p divides (ap-1-1).

Q: Let n ∈ N and p be a prime. Prove that if n is not divisible by p then n and p arerelatively prime

A: From Prime factorization theorem we know that any natural number can be written as product of…

Q: Prove that if p is prime and pla*, then pla, and hence p*|a*; is this still valid if p is composite?

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Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

Q: Let be an odd prime. Prove that for all i with 1 sisp-2 one has p| 1° + 2' + ... + (p – 1)'.

A: Given : p is an odd prime To Prove :          p| 1i+2i+. . .+p-1i for all i with 1≤i≤p-2

Q: Prove for every pair p, q of distinct primes that

A: To prove that for every pair p, q of distinct primes that pq is irrational.

Q: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

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Q: Prove that there are infinitely many primes of the form p = 4n + 3.

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Q: Prove that if D is a PID and p ∈ D is irreducible, then p is a prime of D

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