Since the next prime after 7 is​ 11, the smallest composite number whose prime factors are all greater than​ 7, is ▼     Since this is ▼   less greater than​ 100, there are no ▼   prime composite ​numbers, from 2 to​ 100, whose prime factors are all greater than 7.​ Therefore, ▼   every at least one composite number from 2 to 100 must be a multiple of some prime number less than or equal to 7.

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: The number of factors of prime numbers are None of the mentioned O 30 Depends on the prime number

A: No. Of factor of prime no. Means how many divisor consist of a prime no.

Q: he sum of any two integers of the from 4k + 1isalways: COdd prime Zero even

A: The sum of two consecutive integers can always be written as 2n+1, but it can be written as 4n+1…

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

Q: Any integer greater than 1 that has other positive factors besides 1 and itself is also known as O…

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Q: Prove: in L number psiz 12, for every prime poiz, [p] = [1], [5) , [7) , or [11]

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

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Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

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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…

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Q: If you randomly pick integers from 1 to 200, how many would you have to pick to ensure that at least…

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Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: Find (characterize) all primes p > 3 that satisfy (금) 1.

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Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

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Q: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

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Q: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

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Q: Show that there are infinitely many primes of the form 6k + 5.

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Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

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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: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. In a proof by…

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Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

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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: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

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Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

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Q: Prove, that if 2m + 1 is a prime number, then m must be a power of 2 (m = 2n, for some n ∈ N).

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Q: I know that using a corollary that the congruence x2 ≡ 0 (mod p) that the only solutions are + and -…

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Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: Any two integers are congruent modulo to: even integer odd integer Prime integer not a prime nor…

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Q: Suppose that 3x ≡ 2 (mod 5). Find the remainder when x is divided by 5; in other words, compute x…

A: ANSWER :

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.

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Prove that any integer greater than or equal to 48 can be represented as the sum of nonnegative…

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Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

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Q: The sum of any two integers of the from 4k + 1 is always: Even O prime Zero Odd

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Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

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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: Prove that there are infinitely many prime numbers that are square modulo 11 (trying to prove…

A: To prove: There are infinitely many prime numbers that are square modulo 11.

Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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Q: Prove that the messenger number M19 is a prime hence the integer n = 218 (219 -1) is perfect

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

A: Solution

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

A: Consider the provided question,