Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For the "if" direction, write n = md with a positive integer and use the factorization amd – 1 = (am – 1) x (am(d-1) + am(d-2) + … + am + 1).

Q: Suppose x is a real number. Prove that if x 2, then there is a real number y such that x =…

A:

Q: Let u = (2,6, 4) and 7 = (-1, -2, - 7). Compute the following: ū. v = vu - ||||²

A: Since you have asked a question with multiple parts in a single request we would be answering only…

Q: If a = (4, 0, 4), b = (3, 4,-4), and c = (0, 4, 5), show that a x (b x c) = (a x b) x c. 60i + 96j -…

A: Solution: Step-1                         Given if a = 4, 0, 4 ,b = 3, 4, -4, and c = 0, 4, 5, show…

Q: 1. Find the Laplace transform of the function f(t) = eat[a cos(Bt) + b sin(3t)].

A:

Q: Show that the polynomial 1+x+x² +...+xP-1 . where p is a prime number, is irreducible over the field…

A:

Q: Let →u=⟨6,−4,−6⟩ and →v=⟨−5,−3,−6⟩. Compute the following:

A:

Q: Let a=⟨−3,−2,2⟩ and b=⟨−4,−2,0⟩ Compute:

A: Given that          a = <-3, -2, 2 >    and    b = <-4, -2, 0> Here we have to compute…

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

A: The multiplication cx where x∈ℝ2 is defined as cx=cx1, 0 because x=x1, x2 where x1 and x2 belong to…

Q: Let V be all elements of R2 that have the form (a, a2) where a is a real number. Show that (3,9) is…

A: Let V be all elements of R2 that have the form (a, a2) where a is a real number. We have show that…

Q: a and b are positive integers, show that gcd(a,b) = gcd(a,a+b).  Discrete math questio

A: To prove that gcd(a, b) = gcd(a, a + b) for positive integers a and b, you can use the properties of…

Q: Prove that if a E Z then the only positive divisor of both a and a + 1 is 1.

A:

Q: Suppose (x1,x2) + (Y1,Y2) is defined to be (x1 + Y2 ,x2 + Y1). With the usual multiplication cx = (…

A: It is given that the addition is defined as x1, x2+y1, y2=x1+y2, x2+y1 and the multiplication is…