Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

Q: If a is an algebraic integer satisfying a³ + a + 1 = 0 and ß is an algebraic integer satisfying B² +…

A: Note: There are many ways of proving. We have done by giving the proof more elaborately for the…

Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

A:

Q: 74. Prove that for any real numbers a and b, |b − a|=|a − b|.

A: Given; a and b are any real numbers Consider, L.H.S=b-aWe can write it as,=b+-a=-1-b+a=-1a-b…

Q: 10) Let a and b be positive integers. Prove that (a, b) = a if and only if a divides b.

A:

Q: Prove the following statements: iii. Suppose that a, b, c and d are positive integers, prove that…

A:

Q: (a) Prove that 3n2 +1 is even if and only if n is an odd integer. (b) Suppose A, B,C and D are sets.…

A: The given question has two parts. The solution is explained below.

Q: Provide a proof by contraposition of the statement below. 2 Prove that for any integer a, if a is…

A:

Q: Prove that, nonetheless, for all integers a, if 2 | a and 3 | a, then 6 | a

A:

Q: Prove: For all integers a,b and c, if a|b and a|c, then a|(3b–2c)

A:

Q: Let n be any integer. Prove that 3n - 2 is odd if and only if n is odd. Suppose x and y are any…

A:

Q: Prove the following by proof by contradiction: At least one of the 5 real numbers a, b, c, d, e is…

A: We will use basic properties  of inequality to proof this result by contradiction

Q: Prove that if a > 2 and b are integers, then a † b or a { (b+1).

A: Given that a≥2 and b are integers. We will prove it by contradiction. Let a|b and a|(b+1) then there…

Q: Prove that for all integers a, b, and c if a divides b then for all naturla numbers n a^n divides…

A:

Q: Prove that for all integers a,b which are not both 0, and for all integers d, d|gcd(a,b) if and only…

A:

Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Prove that for each irrational number r, there exists a unique integer n such that |r – n| < 1/2.

A:

Q: Prove that for every real numbers a, b, and c, where b a is not equal to zero, if a +b = c, then=+1

A: Given,            For every real numbers a,b and c, where b is not equal to zero.            if…

Q: Prove that for all the integers a and n, if a | n, then a^2 | n^2

A: Given that a | n. This means n=ak, where k=integer. Then we square both sides: n2=(ak)2 or n2=a2k2

Q: Prove: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

A:

Q: For all integers a, b and c, a|bc if and only if a|b and a|c True or False

A:

Q: ivided by 6.

A: Given, m is odd. So, m = 2k+1 for some integer k  &amp;, n leaves remainder 2 when divided by 3  So,…

Q: Provide a proof by contraposition of the statement below. Prove that for any integer r, if a is odd,…

A:

Q: For every pair of integers x and y, if x – y is odd, then x is odd or y is odd.

A: Let x-is even integer and y-is even integer. x=2m, y=2n

Q: 3. Show that if a and b are integers such that alb, then ak|b* for every positive integer k.

A:

Q: Prove the statement for positive integer n and r, with r ≤ n. ( n n − r ) = ( n r )

A: We know that, the term nr for positive integer n and r, with r ≤ n is given as nr=Crnnr=n!n-r!r!…

Q: Prove, for all integers x, that if x2 is odd, then x is odd.

A: Result: Every odd integer n can be written as n=2k+1, where k is an integer. Every even integer n…

Q: For all integers n>0, prove: 2n < (n+2)!

A:

Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

Q: Let a and b be integers. Proof  by contrapositive that a + b is an even number if and only if both…

A:

Q: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

Q: Prove that if a and b are integers such that a|b, then either a = b or a = -b

A: Take a = 2 ,b = 4 . We have a divides b but neither a = b , nor a = -b.

Q: Prove: if a and b are integers such that a is a factor of b,  then for any integer c, a is a factor…

A:

Q: a. Prove that for every integer a, if a³ is even then a is even. b. Prove that 2 is irrational.

A:

Q: Prove that if a is an integer and b is a positive integer, then exists unique integers q and r with…

A: Please check further steps for solution.

Q: Prove by contradiction that for all integers a, b, and c, if a|b and a | c, then a |(b + c).

A:

Q: Prove that the difference between two integers is odd if and only if exactly one of the integers is…

A: Suppose x be any integer. If x is odd then x=2m+1 for some integer m. If x is even then x=2m for…

Q: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

A:

Q: For any integers, a, b, c, show that if a|b and a|c, then a|(k1b+k2c) for any k1, k2∈Z.

A: Division Algorithm: Let x and y be two integers with y&gt;0.  Then there exist unique integer q and…

Q: Let n, a, and b be positive integers such that ab = n. If (a, b) = 1, prove that there exist…

A:

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Let a and b be any integers. If there exist integers s and t such that as + bt == 2 then a. a and b…

A: SOLUTION-

Q: Let a, b, and c be integers. Prove that if a divides b and a divides c, then a divides (b-c).

A: a, b and c are integers, a divides b and a divides c. Since a divides b, therefore b=ak,  k is any…

Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

A:

Q: Prove that there do not exist positive integers a and n such that a? + 3 = 3".

A:

Q: For all integers a, b, and c, a | bc if and only if a | b and a | c. ect one: True False

A:

Q: Prove the following result.  For all integers a and b, if ab is even, then a or b is even. how…

A:

Q: Prove or disprove: For all integers a and b, if a +b is odd, then a is even and b is odd.

A:

Q: Prove that if a is odd and b is odd, then (a + 1) b is even

A: Number system.. properties

Q: Prove or disprove:  For all integers a and b, if a+b is even, then a is even and b is even.

A: We disprove this statement by taking a counter example.

Q: a) Prove, for all integers k, that 2k-1 is odd. b)prove, for all integers x, if x is odd, then x =…

A: