An integer n is divisible by k e Z when there exists another integer m satisfying n = Integers that divide n are called factors. A proper factor k dividing n satisfies 0 < k < n. A perfect number is an integer n > 0 that can be written as the sum of its proper factors. For example, the number 14 is not perfect: the set of proper factors of 14 are {1,2,7}, but 1+2+7= 107 14. km. Prove the following statements. (a) At least one perfect number exists. (b) For any prime integer p, the positive integer p? is not a perfect prime.

Q: Let n be an integer. Prove that GCD(n² n + 1, n + 1) equals 1 or 3. Show by examples that both…

A:

Q: 10. If j and k are both prime numbers, and j> k, which of the following statements must be true? I.…

A: Algebra problem....

Q: (12) Let p be a prime integer, p≥ 5. Prove that p² - 1 is divisible by 24.

A:

Q: Questions: Confirm that gcd(30,40) * lcm(30,40) = 30 * 40. Find integers a and b such that 30a…

A: The gcd(30,40) is 10 The lcm(30,40) is 120.

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

A:

Q: For any integer ? show that the pair of integers 7? + 9 and 4? + 5 are relatively prime.

A:

Q: Show that for every positive

A:

Q: Show that if p is a prime number, a and b are positive integers and p | ab, then p | a or p | b.

A: The objective is to prove that if p is a prime number, a and b are positive integers and p | ab,…

Q: How do I prove: Let p be a prime number. Show that for any integer a the greatest common divisor of…

A:

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

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

A:

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

A:

Q: Prove that n is odd if and only if 3n+9 is even. You must use definitions of even (and odd, if…

A: An even number is an integer that is divisible by 2, i.e., when divided by 2, the remainder is 0.An…

Q: Decide which of the following are valid proofs of the following statement: Let n be any integer.…

A: Let n be any integer. Then n3-n is divisible by 3.

Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

A:

Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

Q: 3. Show that there exist 100 consecutive positive integers that are not perfect squares.

A: Here we have to show that there exists 100 consecutive positive integer that are not perfect square.…

Q: Denote by an the nth positive integer that is not a perfect square. For example, we have…

A:

Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

A:

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: 2. The Hellenistic mathematician Nicomachus studied the following concept related to perfect…

A: