Consider the following true fact: For all n, m € Z \ {0}, if ged(n, m) = 1 then there are a, b € Z with an + bm Is the converse also true? Prove your claim. = 1.

Q: d) P(-1< Z <1.8)

A: d) P(-1&lt; Z &lt;1.8)

Q: Show that Sn is solvable when n ≤ 4.

A: We see that S1  is the trivial group {e} and it is solvable.   We see that , |S2| = 2 so it is…

Q: 7. Given that , -f() = 15, which of the following must be true? O-35, f (P) dp = 15 O-3 f (p) dp =…

A:

Q: For every positive 2-digit number, x, with tens digit / and units digit u, let y be the 2-digit…

A:

Q: Let U = N, A = {1, 3, 5}, B = {x : x² - 6x + 5 = 0}, and D = {x:x>5}. Find d. A-D e. A x B f. BxB g.…

A:

Q: Suppose that x is N(15,4). Find the P(X>11).

A: The random variable X follows normal distribution. The population mean is 15. The population…

Q: . Show that p(X) = XP-1 +... +X +1 when p is prime.

A: For every integer n≥1, the nth cyclotomic polynomial Φn is the monic polynomial whose complex roots…

Q: (a) Consider the following true fact: For all n, m € Z \ {0}, if ged(n, m) Is the converse also…

A: If Converse of any one of the above is not true,  then we will discuss it with the help of…

Q: 2. (6 marks) Let a, b, c € Z with a 0. Prove that if a X (a + 2b + 3c), then a Xb or a Xc.

A:

Q: What conditions, if any, must be set forth in order for a + b to be equal to n(A U B)?

A:

Q: = (-10,6, –4) be in R and a be in R. Then , 36 , Let x ат — ).

A:

Q: For all n, d EZ where d + 0, d\n n = dk for some kEZ. Disprove: For all integers a, 6, and c, if…

A: I have disproved the result

Q: Show that (a + b)x = ax + bx for all a, b E F and all x € F¹.

A:

Q: Show that E(r(r + 1) =- n(n + 1)(n + 2), n E N.

A: We have to show that ∑r=1n(r(r+1))=13n(n+1)(n+2)           ∀n∈ℕ