If p 2 5 is a prime, then p² + 2 1) prime 2) perfect integer 3) composite (Hint: L et n - 6L

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

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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: By dragging statements from the left column to the right column below, construct a valide proof of…

A: To prove the below statement: For all integer n, if 9n is even, then n is even

Q: 14. Use the indirect method (contrapositive) to show that "If a + b is an odd integer, then a or b…

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Q: Two-digit natural numbers are formed, with replacement, from the digits 0 through 9. How many…

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Q: Want a rigorous proof   Prove that The set of prime numbers has no largest element.

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Q: 10. How many positive integers between 100 and 999 inclusive a) are divisible by 7? b) are not…

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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: 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: 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: If both p and p^2 +8 are prime numbers then show that p^3+4 is also prime number

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Q: If n is an integer and 3n + 2 is even then n is even. (You may use any kind of proof)

A: Given: To prove that if n is an integer and 3n+2 is even then n is even.

Q: 3. Are 3, 33 and 18 pairwise relatively prime? A) True B) False

A: Here we explain about prime numbers

Q: 6. Calculate the number of different six-digit numbers which can be formed using the digits 0,1, 2,…

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Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is…

Q: 1. How many positive integers of 5-digit-number can be formed from the prime numbers starting at 1…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 5-a) How many prime triples of the form (p, 2p + 1, 4p + 1) are there? Describe them all, if…

A: a) The objective is to state the number of prime triplets of the form p,2p+1,4p+1. b) To verify the…

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: 5. Let n be a positive integer and p1,P2,P3, . „Pn be n distinct primes. If m is a positive integer…

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Q: Which of the following 2 numbers are relatively prime? O ged(12, 48) is 12 O gcd(12, 41) is 1 O…

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Q: b) If x is an even prime number, then x is divisible by 2. Converse: Inverse: Contrapositive:

A: Given the statement, "If x is an even prime number, then x is divisible by 2". It is of the form,…

Q: If n is prime, then 2" – 1 is prime.

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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: 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: If a, b, c, d are distinct prime numbers with an as smallest prime then a * b * c dis None of the…

A: Given a,b,c,d are distinct prime numbers with a as smallest prime. Then, a*b*c*d=_________

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: (4) Consider the statement "There exists an even integer n, such that for all rational number q, we…

A: Statement "There exists an even integer n, such that for all rational number q, we have n²>q"

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

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

A: See the handwritten solution

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

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Q: If a, b, c, d are distinct prime numbers with an as smallest prime then a *b * c dis None of the…

A: Answer

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: 4. Let a, b, and d be integers such that dļab. Is it true that either da or d b? If you claim that…

A: Given integers a,b and d. We need to check whether if d/

Q: Given : "If n is an odd integer, then n² is odd.", what do you assume in a proof by contradiction? O…

A: Given that"If n is odd integer , then n^2 is odd." Here we find what should we assume in the proof…

Q: For all odd primes p, show that p 2(p-3)!+1.

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Q: If a, bị c, d are distinct prime numbers with an as smallest prime the a * b * c *d is Prime number…

A: Given: a, b, c, d are distinct prime numbers and  a is the smallest prime number We know that, the…

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

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Q: Choose the correct choice. If none of the choices is correct, choose "Nor Let n be a positive…

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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: Show that if n is an integer and n³ + 5 is odd, then n is even using a) a proof by contraposition.…

A: Proof by contradiction and contraposition

Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural

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

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Q: 4. How many positive natural numbers less than 1000 are neither divisible by 3 nor 5 nor 7 nor 11.

A: Let U={1,2,3,...1000} n(U)=1000 A={a∈U/ x is divisible by 3}A={3,6,9. . .…

Q: Let p2 q be primes greater or equal to 5. Show disprove that 24 divides p² – q². or

A: Consider the provided question,