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

Q: Let i be a positive integer and A, [-i+ 1,1]. Then U14, %3D O R O None of these O F1,11

A: Given that 'i ' be a positive integer and Ai=-i+1,1. We have to find the union ∪i=1∞ Ai.

Q: If the A.M. and G.M. between two numbers are in the ratio mn, then prove that the numbers are in the…

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Q: Let "xy" denote "x does not divide y." Prove by any method. Let a and b be integers. If 3 ab, then…

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Q: Let P and Q be positive odd coprime integers. Prove that P – 1 Q – 1 2 2 m<Q/2 n<P/2

A: Let us consider a rectangle in xy coordinate with vertices 0,0, p2,0, 0,q2, p2,q2 . Now in this…

Q: Given a positive integer n, let a2,n–1 a2,n a3,n-2 a3,n-1 a3,n An = аn-1,2 аn-1,n-2 аn-1,п-1 @n-1,п…

A: Given: Here the given positive integer n, Let An = 00...00a1,n00...0a2,…

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Q: Let A be the set of irrational numbers in [0, 1] and let B be the set of ratinal numbers in [0, 1].…

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Q: a) Find integers a and b such that a + b =a-b (mod 5).

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Q: z = (a + bi)4n+ (b + ai) 4n; a ∈ ℝ ; b ∈ ℝ; n ∈ ℕ. Show that z is a real number.

A: Properties of complex number : If z is real number than z=z¯ . Suppose z=x+i y than z¯=x- iy .…

Q: Prove that's is prime p, then (a + b)² = (a² + b²)(mod p), Va; b € Z

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Q: 5. Let a be a real number, c an integer, c > 0. Prove that

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Q: Show that for integers a, n satisfying 0 < a <n, ged(a, n) = gcd(n, n-a)

A: Solution is given below:

Q: Prove that 1lellse TlI

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Q: are 51 software students taking Let pi, p2,.. Psi be their final cam scores. Hence, cach p, is an…

A: Given that p1, p2, p3,..., p51 are the final exam scores of 51 software students who are taking MATH…

Q: Let m 2q be an even positive integer. Prove that every integer is congruent modulo m to a unique…

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Q: Show that there is exactly one positive number u such that In(u² + 1) + Tu = 2021.

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Q: Construct A so thatN(A) = all multiples of(4,3,2,l). Its rank is __ .

A: We have to construct a matrix A so that NA=all multiples of 4, 3, 2, 1 and also find its rank.

Q: Write down the supremum of the set {7+ 7/n2 − 5/m | n,m ∈ N}. prove ur answer from the definition.

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Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: 10. Let m and n are two positive integers with m > n. Let a be a positive integer. Prove that a2" +1…

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Q: 1. Prove that f1 + f²+...+ ƒ/²/2 = ƒnƒn+1, where n is a positive integer and fn is the nth Fibonacci…

A: The Fibonacci numbers are the numbers in the following integer sequence.…

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

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Q: III) Are there positive integers x and y suct that x³ = 2² +15? (Hint: for every integer x, x³ 0, 1,…

A: We shall show that no such positive integers exist.

Q: Let n be a positive integer. Suppose that there are three pegs and on one of them n rings are…

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Q: let a, b, and m be integer, prove that if 2a+3b>=12m+1, than a>=3m+1, or b>=2m+1

A: To prove: For integer a, b, and m if   2a+3b≥12m+1Than, either a≥3m+1, or b≥2m+1.

Q: Find a number a such that 70a = 1 mod 17.

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Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: Prove that 80

A: Let An=-1n,2+1n and A=∩n=1∞-1n,2+1n=a,b Since, Ai⊃Aj whenever j&lt;i, we can find A by simply taking…

Q: Prove this statement: we have numbers {a1, a2, a3 .,a2n} which are even and positive, but 0 < {a1,…

A: As, we know that, A.M.≥G.M. ---(1) A.M. of numbers, a1,a2,a3...a2n=a1+a2+a3+a4...+a2nn G.M. of…

Q: Let a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: p, q, pq | (apq − ap − aq  + a) for all a.

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Q: Let а, b and c be integers. Use the method of proof by contradiction to prove that if a|b and a f c,…

A: We have to solve given problem:

Q: If P, stands for "P., then prove that I+1. P, +2. P, +3. P, +...+n. P, = (n+1)!.

A: that is will be...equal the 1 + 1. 1P1 + 2. 2P2 + 3. 3P3 + …. n. nPn = n+1Pn+1. and we know that nPr…

Q: Show that there is exactly one real number t such that eTt + te" 207 + V2021.

A: Let us take an equation: g(x)=t Where g(x) be any function of x In order to show existence of x such…

Q: Let {\mathcal S}S be the set of all (nonnegative) 4-digit integers. How many integers does {…

A: Here given S is set of all (nonnegative) 4-digit integers We have to calculate,  How many 4 digits…

Q: Let A, BeM,(C). Prove that for every integer k21, tr(AB) =tr(BA) Does tr(AB) =tr(A*B* )? Explain…

A: We will do both these subparts of this question for you. Solving this question requires basic…