Let a, b be integers. If ab is even, then at least one of a or b is even.

Q: If a, b. and c are integers such that a|b and c|b. then ac|b.

A: False

Q: Let a,b be integers, not both 0. Show { na + mb : n,m ​​​​​are integers} = iff a and b are…

A: We have to prove the given statement.

Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

Q: If a, b, and c are integers such that a|b and c|b. then ac|b. ylg

A:

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

A:

Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

A:

Q: 6. Prove that if x, y, and z are integers and x + y + z is odd, then at least one of x, y, and z is…

A: Given: x , y , and z are integers and x + y + z is odd.  We have to show that at least one of x , y…

Q: Let a, b, c be positive integers and let d = (a, b). Since d|a, there is an integer h with a = dh.…

A:

Q: If a, b, and c are integers such that b is a multiple of a³ and c is a multiple of b², then c is a…

A:

Q: For all integers x and y, if 6x+y is odd, then x and y are both odd. True False

A:

Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

A:

Q: Let x and y be integers. Prove that if x2 + y2 is odd, then x+y is odd

A: Given x and y be an integers. We have to prove that if x2+y2 is odd then x+y is odd. Since given…

Q: Show that if a, b, c, and d are integers with a and c nonzero, such that a|b and c|d, then ac|bd.

A:

Q: 1. Let a, b, and c be integers, where a +0. Prove or disprove that if a | b and a c, then a | ( b -…

A:

Q: If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that

A:

Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

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: Prove that If k, q, w and r are integers, and if c|k and c|q, then c|(kw + qr)

A: We know that  Definition of Divisibility : If a and b are integers such that a≠0, then we say "a…

Q: If a is any integer, is a(a + 1) even or odd?

A:

Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

A:

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

A: " Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: If a, and c are integers such that albc, then alb or alc. :Select one True False

A: Given integers 'a' 'b' and 'c' such that a|bc then a|b or a|c. This statement is false.  Let us…

Q: Let a and b be integers,not both zero .Then a and b are relatively prime if and only if

A: Solution are as follows

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

A:

Q: If a, b. and c are integers such that alb and c|b. then ac|b.

A:

Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

A:

Q: If a and b are integers with gcd(a, b) = d Then gcd (-- d.

A: Given a statement

Q: Suppose that a, a', b, b' are positive integers satisfying a) ab' a'b = b) gcd(a, b) = gcd(a', b') =…

A: Given that a,a',b,b'are positive integers such that  ab'=a'bg.c.da,b=g.c.da',b'=1 We have to show…

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

A:

Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

Q: (1) Prove or disprove that if a, b, and e are integers and a + -2, then at least one of a and b is…

A: The complete solution is in given below

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

A: Suppose a|b and a|c. Then there are integers m and n such thatb = am and c = an.Then b + c = am + an…

Q: If there exist integers r and s such that ra + sb = d. then d = (a, b). hi O

A: Given: There exist a two integers r and s such that ra+sb=d. To prove : d=(a,b)

Q: If a, b, and c are integers such that b is a multiple of a and e is a multiple of b², then c is a…

A:

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

A:

Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

Q: If a, b, and c are integers such that alb and c|b. then ac|b. ylgn O ihi O

A: In order to prove this statement, we first need to understand what the math notation a∣b implies. To…

Q: Let, a1 = 3, a2 = 4 and for n 2 3, an = 2an-1 +an-2 – 5, express an in terms of n.

A: We will first find the roots of characteristic equation and write general solution of homogeneous…

Q: 2. Given integers a, b, c, d, verify the follOWilng. (a) If a | b, then a | bc. (b) If a | b and a |…

A: Note: according to our guidelines we can answer only first three subparts Given integers a, b, c, d,…

Q: Let m, n, p be integers. Use direct proof to show that if m+n and n+p are odd then m+p is even.

A:

Q: Let n and r be non-negative integers such that r <n. Then, C + "C , = **C,

A: nCr=n!r!(n-r)!

Q: Prove that if m and n are integers and mn is even, then m is even or n is even.

A:

Q: If there exist integers r and s such that ra + sb = d. then d = (a, b). %3D

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 that if the product of two integers ab is odd, then a2 + b2 is even

A:

Q: If a and b are two integers such that a = bq +r, with b> 0, then %3D gcd(a, b) # gcd(b,r) gcd(a, b)…

A: Let a = bq + r   … (1)Now,  let d1   = gcd ( a,b) then ⇒d1  | a  and   d1  | b ⇒d1 | a - bq   and d1…

Q: Prove that if p,q, and r are integers, r|p and r|q, then r^2|pq.

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: Show that if a | b and b | a, where a and b are integers, then a = b or a = −b.

A: We show that if a | b and b | a, where a and b are integers, then a = b or a = −b.