If p is a prime number greater than 3, which of the following represents the sum of the positive prime factors of 3p?

Q: 21. Find two different factorizations of the number 21 into primes of the form a + b v-2, where a…

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Q: If b is a positive integer not divisible by the prime p, then (р-1)b +(÷) +(÷)- 2b 3b %3D ... 1 O -1…

A: If b is positive integer not divisible by the prime p, then

Q: An integer n > 2 is composite if and only if n doe

A: Given Statement: An integer n > 2 is composite if and only if n does not divide (n - 1)! + 1.…

Q: The number of factors of prime numbers are Depends on the prime number 3 O None of the mentioned O

A: Prime numbers are the numbers which have only two factors one factor is 1 and other factor is 2

Q: If a and b are two distinct prime numbers, then the greatest common divisor of a and b is O ab

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Q: 2. If b and c are relatively prime and a / bc then a / b or a /c.

A: First Method: Since b and c are prime: gcd (b,c) =1 Then: gcd (ab, ac) is equal to "a" Then: Let for…

Q: Let p > 5 be the prime number. Prove that the expression p – 10p +9 is divisible by 1920.

A: We have to prove that the expression p4-10p2+9 is divisible by 1920.

Q: If n has a prime factorization n = pi*p2² ... p" then the sum Eajn d³P(d) is equal to: a1. az

A: Euler's phi function is widely used in number theory and algebra. This function represents the…

Q: If a, b are two distinct prime number than a highest common factor of a, b is: (2 ab

A: HCF of two numbers is found out by taking common prime factors from both the numbers the…

Q: 9. Which of the following prime numbers is not a factor of the integer 330? (1) 11 (3) 3 (2) 7 (4) 5

A: to select not factor of 330

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: 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 > 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 > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: The integers 77 and 105 are relatively prime.

A: Relatively Prime integers: Two integers mand n are called relatively prime if gcd(m,n)=1. Given two…

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: 1. Consider the two-digit positive integers 10, 11, 12, .., 99. Answer the following questions. a.…

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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: . Which of the following prime numbers is not a factor of the integer 330? (1) 11 (3) 3 (2) 7 (4) 5…

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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: Do you thinkthat there are infinitely many primes of the n -1

A: there are infinitely many primes of the n -1 Explain the above statement

Q: Show that if m is a positive integer, then p(m²) = m· ¢(m). Hint: consider the prime factorization…

A: Introduction: The phi function of Euler denoted by ϕ is one of the most important number-theoretic…

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

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Q: If n = p,P2 ... P where the p, are distinct prime numbers, how many ways can n be written as a…

A: First let us formulate the problem in a different way. Look! we have to find the total number of…

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: 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: integers a, b, and c with a≠0, if a∣b,  then a∣(bc). I need a know show table and a proof

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Q: Suppose “a” is an integer. If 5|2a, then 5|a.

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Q: If n is a product of twin primes, then a) Ø(n)o(n) = (n – 1)(n – 3) b) ø(n)o(n) = (n + 1)(n – 3) %3D…

A: If n is a product of twin primes, then a)  ϕ(n)σ(n) = (n+1)(n-3) b)  ϕ(n)σ(n) = (n+1)(n-3) c)…

Q: describe (a,p)

A: Given, if p is a prime, and a is a positive integer.

Q: If o(n) = p + 1 for some prime p, then n = p. Select one: True False

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Q: Suppose that 56 % of students play football, 53 % of students play cricket, and 12 % of students…

A: Let F=students play footballC=students play cricketGiven,P(F)=0.56P(C)=0.53P(F¯∩C¯)=0.12

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: Give an example of a prime p of each of the forms: 6k + 1 6k + 5.

A: We need to give an example of a prime p of each of the given forms. Let us take k=1. So, we get…

Q: Given that 7,999,999,999 has at most two prime factors, find its largest prime factor.

A: 7,999,999,999 has at most two prime factors, We have to find its largest prime factor.

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: Use trial-and-error to determine the smallest integer n for which the given statement is true. 6n <…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let p > 5 be the prime number. Prove that the expression p* - 10p? + 9 is divisible by 1920.

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Q: 25. If p>1 is an integer and nłp for each integer n for which 2<n< p, then p is prime.

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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: 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

Q: Find all values of primes p for which (¹¹) = 1 Р

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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: If X is a positive integer divisible by 6, and X <60 What is the greatcst possibe vaive of X 7

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

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Q: if n zo is an integer, then n!=n(n-)3.21 ...

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Q: Let N be an integer, and let p be a prime > 7. Suppose p| (N2+7). Show that p is of the form 1+ 7k,…

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Q: Let M= (1 x 2 x 3 x 4 x ... × 1000) + 1 What can you say about the prime factors of M? Why?

A: M= (1 x 2 x 3 x 4 x ... × 1000) + 1 Prime factor of this number is the number itself, there is no…