If you want to check that 613 is a prime number using the test for primality, what is the minimum amount of numbers you must divide into 613 (or use a divisibility rule) to determine this?

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: How might you tell if a number is even or odd in bases two, three, four, five, six, seven, eight,…

A: Given data:- The given numbers are 2, 3, 4, 5, 6, 7, 8, 9, 10 Explain the generalize to bases of n.

Q: (1) What is gcd(31212021 + 3, 31212021 - 3)? I am aware of 3121 is a prime number, and 2021 is…

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

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: What is the converse of: an integer has divisors other than 1 and itself, if an integer is…

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Q: Find the prime factorization of 793169. To show your work you must use Theorem 2

A: Consider the equation

Q: The number of factors of prime numbers are Depends on the prime number None of the mentioned (3 2

A: Given query to find number of factors of a prime number.

Q: What is the smallest positive integer n such that 80325 divides n!

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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: Determine the number of natural numbers less than 700 that can be written using the digtits 5,6,7,9?

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Q: Determine the number of integers between 1 and 10000 that are not divisible by 6, but are divisible…

A: Given, the number of integers between 1 and 10000 To find the integers between 1 and 10000 that are…

Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: consider to test whether 9010 is​ prime?

A: We have, √9010 = 94.9210198 So, the greatest prime which is less than 94.9210198 is, 89 So, by the…

Q: A number N is divisible by 4 iff the number formed by the last 2 digits of N is divisible by 4. For…

A: We can find number divisible by 4  by using the concept given in the question

Q: The number of factors of prime numberss are (None of the mentioned Depends on the prime number (2 (3

A: Prime number is only divisible by 1 and that number itself

Q: Find the Prime Factorization of 648, 1056, and 1496. Provide at least one divisibility test for each…

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Q: Every natural number greater than 1 can be written as the products of prime numbers.

A: Prove that  every natural number greater than 1 can be expressed as product of primes or the number…

Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: The Great est Common Divisor of two prime numbers a and b is

A: Given : a and b are two prime numbers To find : Greatest Common Divisor of a and b.

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: At the end of 2005, the largest known prime number was 2^30402457 -1, write an expression for the…

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Q: What is the smallest positive integer that has 6, 8, and 10 as factors?

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Q: 16. Devise a way of checking whether or not a number is divisible by 24. Test your method on 36 and…

A: 16

Q: The number of positive integers not exceeding 1000 that are not divisible by 8 and not divisible by…

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Q: There exist two prime numbers and such that their product gives 1261

A: We have to find two prime numbers whose product gives the required number

Q: noosing a positive integer from 100 to 999, what is the poSSibility of choosing a positive integer…

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Q: How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

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Q: Find an integer between 100 and 150 divisible by 9.   How do i figure this out, I am completely…

A: A number is divisible by 9 if the sum of the digits is divisible by 9.

Q: Write the set of natural numbers greater than 16 using the most concise method.

A: We need to write the set of natural numbers greater than 16 using most concise method. We know that…

Q: Show that n-lis divisible by 8, if n is an odd positive integer

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Q: Every odd prime number is greater than 2.

A: Given: "Every odd prime number is greater than 2." To find: Statement in the form "If p, then q"…

Q: Show that one of the prime divisors of any three-digit composite number is less than 31 prime, or…

A: A composite integer a has a prime factor p only if p≤a that is if p≥a than there exist another prime…

Q: Describe a general method for finding all the divisors of a positive integer once you know the prime…

A: Given  A positive Integer whose prime factorization is given . To find all the divisors with general…

Q: 21 and 24 are not relatively prime numbers

A: To find: Whether 21 and 24 are relatively prime numbers or not?

Q: Find the sum of all positive integers, from 5 to 1555 inclusive, that are divisible by 5.

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Q: What is the greatest common divisor of 24 and 44? n If n is a natural number such that gcd(800, n) =…

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Q: 5. Find the number of positive integers less than 700 that are divisible by 3 but not divisible by…

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Q: Explain why it is convenient to write the expression for the number of possible permutations in…

A: It is efficient and time saving, imagine if you want to write 4! in its entire multiplication form…

Q: the greatest prime you must consider to test whether 3524 is prime?

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Q: Determine the possibilities for the final digit of a sixth power of an integer.

A: The last digits of a number are 0,1,2,3,4,5,6,7,8 and 9. The number 0 raised to any number will be 0…

Q: 3. There are seven students who always compare their CS test grades. They discovered that either the…

A: It is given that there are seven students who always compare their CS test grades. They discovered…

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: DOference of twin primes is not divisible by 2.

A: We have to find that the difference between the twin primes are divisible by 2 or not

Q: How many twin primes are there where both primes are less than or equal to 31? Number of twin…

A: Twin prime: If there is only one composite number between two prime numbers, then that two numbers…

Q: Show that in any collection of 7 or more natural numbers there are ALWAYS TWO numbers whose sum or…

A: In the given problem we have to show that in any collection of 7 or more natural numbers there are…

Q: Find the sum of the divisors of 120. Generalise the result for an arbitrary natural number M

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Q: How many corresponding integers are made up of 1443 digits taken from (1-2-3-4-5-6) provided that no…

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Q: 1.What is the smallest number that can be exactly divided by 2, 3, 4, 5  and 8 ? 2.What are the…

A: To answer the given questions.