x and y. Prove that if alb and alc, then a|(xb+ yc) for any integers

Q: Let a, n E N. Assume that n > 2. Show that if Va is rational, then it is an integer.

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Q: Verify that for all x, the change-of-base formula holds: bx = ax Ioga b for a, b > 0 such that a…

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Q: Icm(a, b). Prove that Let a and b be positive integers, and let d = Za × Z, = Za × Zm· gcd(a, b) and…

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Q: Prove the following: If ā and b are in R', then (5 -a)×(5 +a)=-2(āxb).

A: Given a⇀ and b⇀ are in R3. To prove :  b⇀-a⇀×b⇀+a⇀=-2a⇀×b⇀

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A: we have 15a+20b =49 and a and b both are in R  we prove that a and b are not both integers.

Q: where z is the set of integers, for X ∈ ℤ− ve Y ∈ ℤ+  prove these; a) if X^2 = 1ℤ X = −1ℤ b) X −…

A: It is given that,   X∈ℤ-, Y∈ℤ+

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x+ 5y² is even

A: Suppose x and y are integers. Prove that if x and y are odd, then x+5y2 is even.

Q: Let an = an-1+ an-2 with ao = 0, a1 = 1. Show that an is even iff ( if and only if) n is a multiple…

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Q: Let a, b eN. Prove that if a + b is even, then there exists nonnegative integers x and y such hat x²…

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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: If m and n are positive integers and w = g(x"y"), then show that dw dw = my dy

A: Given w=g(xmyn).

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

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

Q: 18. Let xo < x, < x2, let {yo, y, y2} be given. Find the polynomial q(x) with deg(q) < 2 and such…

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Q: Determine a and b such thatp(x) = x2 +ax+ bsatisfies p(l) = 0 and p' (1) = 4.

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Q: Let a, b € Z and both nonzero. Prove that gcd(a, b) x lcm(a, b) = |ab|.

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Q: Let a and b be two integers. Prove that if a+b is even then a-b is even.

A: Given that a and b are any two integers. We are trying to prove "if a+b is even then a-b is even."

Q: Let a, b and m be integers. Prove that if 2a + 3b ≥ 12m + 1, then a ≥ 3m + 1 or b ≥ 3m + 1.

A: Here I have corrected the statement of the Question and provide a detailed solution of the problem.

Q: 4. Let R be the set of all real numbers, xER and n EN such that n 2 2. If (1+x) > 0, prove that Pn:…

A: Let ℝ be the set of real numbers, x∈ℝ and n∈ℕ such that n≥2. If 1+x&gt;0, we want to prove that  Pn…

Q: If all entries of A and A -1 are integers, prove that &lt;let A = 1 or -1. Hint: What is det A times…

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Q: Given x > 0 and a real number y, there is an n in N such that nx > y.

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Q: Prove that there are integers x and y such that 10x-13y=1

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Q: Let a, y E R. Prove that if 21a + 35y = 45, then a and y are not both integers.

A: An integer is colloquially defined as a number that can be written without a fractional component.…

Q: Prove that for any integer a, [a]=m = [r]=m where r is the unique remainder when a is divided by m,…

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Q: 1. (a) Prove that if alb and alc. then alb² + 10c + a.

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Q: 4. Show that if x, y, z are integers such that x2 + y2 + z? = 2xyz, then x = y = z = 0. (H)

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Q: Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such…

A: Given K be any field and a1,a2,...,an be pairwise distinct elements of K. For each i=1,2,3,...,n, pi…

Q: Let {x1, x2,..., xn} C R be a subset of real numbers. Prove that (x1 + x2 + . + an)² < n(x² + x +·.…

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Q: Let n E N coprime with p. Show that there exists x, y E Z coprime and such that p | x² + ny² is and…

A: To prove the given claim we have to show that :- x,y ∈ Z ; gcd(x,y)=1 ; p | x2 + ny2   implies…

Q: For each subfield S of Q[i, z], determine Auts(Q[i, z]).

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Q: Represent algebraically and graphically the set of values of z for which |z+3| = |2 – 2i|.

A: Solution

Q: If x and y are real numbers and x>1, prove that there exists a positive integer n such that x" > y

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Q: Let a be any positive number. Prove that |x| &gt;a if and only if x &gt; a or x &lt; -a.

A: Suppose

Q: 4. Demonstrate that for positive integers t and Q, (t <= [V@]) is equivalent to (t x t <= Q). Show…

A: Given that t and Q are two positive integers and t≤Q To show that t×t≤Q Since from the given…

Q: If a<b<c<d, then prove that for any real 2,

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Q: Represent algebraically and graphically the set of values of z for which |z+3|=|z−2i|

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Q: Given that * :Zx Z Z is defined by m * n = m" for m, n e Z, then is an operation on the set of…

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Q: a. Let, a and b be integers such that 1 ≤ a ≤ b. If b = aq + r, where 0 ≤ r ≤ a, then prove that,…

A: What is Greatest Common Divisor: For two integers, their greatest common divisor is an integer that…

Q: If z = a+bi for some real numbers a and b, show that |2| = |5|.

A: In the regular geometry on R3, we can compute the distance between                 x,y∈R2 and the…

Q: Let, a1 = 3 and for n > 2, an = 2an-1 +1, express an in terms of n. %3D

A: We will find out the required expression in the term of n.

Q: let a>0. prove that if x is a number such that |x-a| a/2

A: Given a&gt;0 and x is a number such that x-a&lt;a2 We will use the property If y&lt;b , then…

Q: Prove that the following identity: (x²+a')' =(x* – 6x°a* +a*) +(4x°a - 4xa³)} 2.2

A: In this question, we had to prove that, (x2 + a2)4 = (x4 - 6x2a2 + a4)2 + (4x3a - 4xa3)2

Q: Prove that for integers a and b, (a -3)b2 is even if and only if a is odd or b is even.

A: We need to prove that for integers a and b, (a -3)b^2 is even if and only if a is odd or b is even.

Q: Prove that for any integer a, [a]=m = [r]=m where r is the unique remainder when a is divided by m,…

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Q: For each xe X and a nbd: U(x) of x there exists another nbd.

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Q: Prove that if a and b are integers, with b > 0, then there exist unique integers q and r satisfying…

A: To Prove -  If 'a' and 'b' are integers, with b&gt;0, then there exists unique integers 'q' and 'r'…

Q: Let A  = {x ∈ A  |x =3b +2 for some integer b } and B = {y ∈ B |y = a + 2 for some integer a }  a.…

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Q: and

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Q: Let x and y be coprime positive integers. Prove gcd(x + y, x² + y²) E {1, 2}.

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Q: If P(1) = 0 and ф(х) >P(x) for all x > 1, then prove that P(x)>0 for all x>1.

A: P1=0 and dPxdx&gt;Px for all x≥1 We have to prove Px&gt;0 for all x&gt;1

Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

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