Prove the following statement directly from the definitions. The product of any two even integers is a multiple of 4. vr such that m =---Select--- v, and there exists ---Select-- Proof: Let m and n be any two even integers. By definition of even there exists ---Select--- such that n = ---Select- v By substitution, mn = 4 - which is an integer because --Select--- v of integers are integers. Thus mn = 4- (an integer), and so 4|mn by definition of divisibility.

Q: Prove that (-1ti3)" + (-1-is3)" has either the value 2n+' or the value -2" if r is any ). integer…

A:

Q: find a Positive integern for which the equality holds %3D

A:

Q: rove that the negative of any even integer is even

A: Assume 2n to be an even integer. 2n = even integer -2n = negative of even integer  Let a = -2n be an…

Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

A:

Q: Prove a^2 +4a+3 is odd if and only if a is even.

A:

Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

A:

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

A:

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

A:

Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

A: Given: Let n be a positive integer such that 78n-1. To prove: 78n+1-1

Q: Prove that among 7 positive integers, there are always two integers such that either their sun or…

A:

Q: Show that 2 + V5 is imational number

A:

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: Show that there are infinitely many examples of two consecutive positive integers that cannot be…

A: According to the given information, It is required to prove that there are infinitely many examples…

Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: If A= xe Z:2" = 1 and B = {x e Z:-4 < 3x-1<11}, then find n(A×B),where Z is a set of integers.

A:

Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

A:

Q: Suppose that x and y represent the amounts of two basic inputs for a production process and that the…

A:

Q: 13. Pier Fermat (1601 – 1665) conjectured that any number F, of the form 2²" + 1 where n is…

A: Fermat number is Fn=22n+1

Q: 6. For an arbitrary integer a, verify the following: (a) 2 a(a + 1), and 3 a(a + 1)(a + 2). 4(b)…

A: a is an arbitrary integer.

Q: is there integer that has reminder of 2 when its devided by 5?

A: There are infinite number of integers that has reminder of 2 when its divided by 5

Q: Prove by contradiction that 2020 cannot be written as a sum of 3 odd integers.

A: The objective is to prove  that 2020 cannot be written as a sum of 3 odd integers by the…

Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

A:

Q: Suppose that there exist integers à and y such that ax + by = 12. What can you deduce about (a, b)?

A:

Q: Prove 8^ +l is composic for any poSitive integer n.

A:

Q: Give a proof by contradiction that there does not exist an integer which is both even and odd.…

A: Assume an integer x which is both odd and even.

Q: Show that det(A) is (a* – b*)/(a – b), where a +b ab A = 1 a +b ab 1 a +b for a and b any real…

A: Given, A=a+bab01a+bab01a+b