Prove that for all integers a, b, and c, with a ̸= 0, if a|b and a|c, then a|(bx + cy)

Prove each statement using a proof by exhaustion.

Let Express in terms of 1 A = [₂ 2 and 2-² = [2] Α B-¹ 3 and find all its entries. ((BA)T)-¹ A and B-¹

N.png ... If gcd(a, m) = 1, then the unique solution xo to ax = b (mod m) is equal to (1) xo = ab (mod m) (2) xo = ab (mod m), where b is the inverse of b. (3) xo = ām (mod m), where ā is the inverse of a. (4) xo = āb (mod m), where ā is the inverse of a. A Choice (2) (B) Choice (1) Choice (4) (D) Choice (3)

Formally prove that on? + 10n + 10 = (n²) using the definition of 2(:).

In this group of problems, you are given the predicate P(x), where the domain of x is the set of natural numbers.

Show that if f (n) and g(n) are monotonically increasing functions, then so are the functions f (n) / C g(n) and f g(n) and if f (n) and g(n) are in addition non-negative, then f (n)/g(n) is monotonically increasing.