2. Let m, n, a, b € Z. Prove that if 6 | m and 6 | n then 3 | (am + bn +15).

Q: Evaluate f1 dy Vy-1 O a. 2ỹ+ 2ln(ỹ – 1) +C O b. 2j – 2ln(/ – 1) + C O c. None\ of\ the\ following O…

A:

Q: Let H = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f} Let W = {(a, b) : a € H, b = H} be the…

A: Answer:-

Q: 8. Let a₁ ≥ a2 ≥ ... ≥ an and b₁,b2,..., bn be two sequences of non-negative real numbers such that…

A: The given problem is to prove the given inequality for the given two sequences an and bn and…

Q: Let U = {1, 2, 3, 4, 3, G, 7, 0, 9, to ; A = { positive even Factors OF to; {1,3,8,9} and {x Ix ic a…

A:

Q: Let a, b, x, y ∈Zbe such that ax+by= 11. May we conclude that gcd(a, b) = 11?Why or why not? (Write…

A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

Q: Let A= {1,2,x,y,z}, and B= {1,2,3,z}. How many distinct elements are in (A*B) intersect (B*A

A: Given that A=1,2,x,y,zB=1,2,3,z To find the number of distinct elements in A×B∩B×A:

Q: 4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b)…

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Q: Determine the least upper bou You must fully justify your an

A: We have to find least upper bound of the given set

Q: |Let A¡ = [i − 1, i] for all integers i = 1,2,3,4. Find the following. a. A₁. A₂- b. A₁ U A₂ U A3 U…

A: The given set Ai=i-1, i for all integers i=1, 2, 3, 4. We have to find, a) A1, A2, A3, A4 b)…

Q: Suppose B C A, B + A, and B ~ A. Prove that A is infinite.

A:

Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

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…

A:

Q: 12. Prove that if d divides a and d divides b, then d divides ax + by for all integers a and y.

A:

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 ImA = A = Aln if A is m X n

A: Given: A be m*n matrix

Q: Prove that if m | k then 2" – 1| 2k – 1

A: Use the definition of Divisibility, "a divides b" means a and b are integers and there is an integer…

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

A:

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: 1. Let a, b € Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 46.

A:

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: Let α belong to Sn. Prove that |α| divides n!

A: The symmetric group ​ is the group of permutations on n objects. Its order is .

Q: 1 Show that n2 + 1 IN V

A:

Q: Let u, v E Z. Show using only the definition of a divisor and algebraic properties that if 7|u and…

A:

Q: A = ({0} U {x |x € Z* and x < 4}), В 3 30, 1, 3, 5, 7}, and C = ({0} U{x| x E Z* and x is even and x…

A:

Q: Show that yn for all n=1, 2₁- ✓

A:

Q: 9. Let a, b, and d be positive integers. Prove that if (a, b) = 1 and d divides a, then (d, b) = 1.

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Q: The greatest common divisor c, of a and b, denoted as c = gcd(a,b), is the largest number that…

A: Here is the solution

Q: 1. The greatest common divisor c, of a and b, denoted as c = gcd(a, b), is the largest number that…

A:

Q: 8. Let K be an integer that is greater than 1, a perfect square, and equal to Ei for some integer D.…

A: A perfect square is a positive integer whose square root is a positive integer. Example of perfect…

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: Let n E Z. Prove that 3 n if and only if 3 | (n² - 1).

A:

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: 1. List a pair of zero divisors of (M₂(Z), +, ×):

A:

Q: Find the centroid of the region. NOTE: Enter the exact answers. (T, 9) (?, ?) %3D y = x7

A: We need to find the centroid of the given region. For finding the centroid we will use following…

Q: Let a1, a2, a3, . .., an be real numbers. Prove that at least one of them is greater than the…

A: We prove it by contradiction.