Prove that if p, q, and r are three different prime numbers, then p² +q² # r².

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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…

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A: The objective is to prove that if p is a prime number, a and b are positive integers and p | ab,…

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

Q: Justify that every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

A: Let ‘P ‘be any prime number Divide ‘p’ by 6, we get P=6q+r       - (1);     r=1,2,3,4,5 And q be any…

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: - Let a, b, n be positive integers. Prove that if a = b (mod n), then (a, n) = (b, n).

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Q: How many integers in the set {1,2,3,4, ..., 200} is divisible by at least one of the primes 2, 5 and…

A: Just use inclusion-exclusion principle for three sets.

Q: how many pairs of primes p and q are there such that q-p=3? Prove your answer.

A: Here, we need to find all the possible pairs of prime numbers p and q such that the difference (q-p)…

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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 it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

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Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.

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Q: If p, q, r are prime numbers such that pqr = 19(p + q + r) then the value of 2(p + q + r) is

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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: Find all numbers p such that all six numbers p, p+2, p+6, p+8, p+12 and p+14 are primes.

A: For the given statement

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A: The details are given in step 2. The idea is to use a certain properties of primes.