1 be rati onal.Then, there exist positive co - primes a and b such that a If possible, let b

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

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Q: Let a,b be integers, not both 0. Show { na + mb : n,m ​​​​​are integers} = iff a and b are…

A: We have to prove the given statement.

Q: Suppose p is prime and that a,b EN with a, b > 1. (a) Show that if p|ak for some k E N, then p|a.…

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Q: 7) Let p be an odd prime, find if a) p =1 mod 8

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Q: Given p > 2 a prime number, prove that: (p+1 |("") = 0(modp 5.1) for any natural number 2 snsp-1 ) =…

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Q: Let p ≥ 2 be a prime. Suppose p does not divide a. Then ap−1 ≡ 1 (mod p).

A: The given problem is to show the given following modular congreunce that let p is a prime and p…

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

A:

Q: 1. Prove or disprove. if alb and bla, where a and b are integers, then a = b.

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Q: Let n ≠ 0 and assume nZ is prime. Prove that n is prime.

A: Given n not equal to 0 and nZ is prime. We prove that n is prime. Let n be a composite number. Then…

Q: If p is a prime with p = 5 mod 6, then (²) O -1 O None of these

A: If p≥3 is an odd prime, then    (-3/p) = 1 if p ≡ 1 (mod 6)    (-3/p) = -1 if p ≡ 5 (mod 6)

Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

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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).

A:

Q: show

A: Given : m+n and n+p are odd  So, we can write, m+n = 2s+1 and, n+p = 2t+1, where, s & t are…

Q: Let r be a primitive root of the prime p with p = 1 (mod 4). Show that -r is also a primitive root…

A: Solution

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: 6. Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Assuming Wilson's Theorem, prove that if p prime and (a, p) = 1, then aP-1 = 1 mod p.

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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…

Q: For any distinct primes p, q,r and s, we have pq # rs. For any positive integer n, there exists n…

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Q: Number theory.

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

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Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p &gt; 5. p2−1≡0mod24

Q: 24. Show that 1P + 2P + 3P + .+ (p - 1)P =0 (mod p) when p is an odd prime. Show that if nis prime…

A: Using Wilson's Theorem, Fermat's Little Theorem and The Pollard Factorization Method, we have to…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

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

A:

Q: If p is an odd prime, prove that: 1² · 32 . 52 ... (p – 2) = (– 1)P*1)/2 (mod p) and 22 . 42 . 62…

A: Given that p is odd prime. (i) To prove 12.32.52....(p-2)2≡-1p+12modp Since,…

Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

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

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Q: Let p be an odd prime number. Prove that [(3)=1] → [p = 1 mod 6).

A: Introduction: An integer-based arithmetic system that takes the remainder into account is called…

Q: Let p and q be distinct primes with p<q and q# 1mod(p). Show that if G|= pq then G Z/pqZ.

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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 and b are primitive roots modulo a prime prime p, then ab is a primitive root modulo p. O True…

A: Answer  ; False Explanation

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)…

Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: 9. Ifn is a squarc-frce integer, prove that r(n) = 2" , where r is the number of prime divisors of…

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Q: Let c = 2^100 and p be the smallest prime such that p >= c. How many composite numbers are between c…

A: Let c=2100 And p=min p is prime:p≥c,p∈ℤ Claim: Total number of composite number between c &amp; p,…

Q: Prove that Let p be a prime. Then (p−1)! ≡ −1 (p)

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Q: Show that if n is prime and b is a positive integer with n ̸∣ b, then n passes Miller’s test to the…

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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: 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 p and q be odd primes and let m and n be positive integers. Explain why U(p") O U(q") is not…

A: given, let p and q be odd primes. and n be a positive integer.

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: Let m, n, p be integers. Use direct proof to show that if m+n and n+p are odd then m+p is even.

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

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Q: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). Show that −r is also a primitive root…

A: The given problem is to show that -r is the primitive root of p when r is a primitive root of prime…

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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Q: prove that, it ņ is an 3n+2is oald then integer and nis add

A: