4. Let a and b be integers. (a) Prove that [a]n = [b]n if and only if a = b (mod n). (b) Prove that either [a]n[b]n = Ø or [a]n = [b]n.

Q: Show that the square of any positive Integer is of the form 3m or 3m+1 for some integer m

A:

Q: Fill in each blank so that the resulting statement is true.

A: We have, 3x=π4+nπ  for any integer n.

Q: Show that the square of any integer is of the form 3m or 3m+1 but not of form 3m+2.

A: Proof :-

Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

A:

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

A:

Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

A:

Q: Let B = [-1 1 4 -3] Find -4B

A: Solution

Q: at a, b and c be real numbers such that a #0 and ac+bc+c 4ac.

A: Given data is : a , b and c are real number such that a≠0 and ac+bc+c2<0 Show that b2>4ac

Q: Let A be a set of n real numbers. Show that A+ A- A € (n).

A: As per the question we are given a set of n real numbers A and we have to show that : |A + A•A| =…

Q: 2. Find the Laplace transform of the function g(t) = (1+t+t + t³ + + tm-1 +tm)et.

A:

Q: (a) f '(x) = 0 X0 O x1 O x2 O x3 O x4 O none of these f "(x) = 0 (b) O xo O x1 O x2 O X3 O X4 O none…

A: To solve the following problem

Q: prove the following statement.

A:

Q: Evaluate or indicate that the root is not a real number √144 + √25 ?

A: To evaluate: 144+25

Q: For positive integers a and b, show that a ab

A: Consider the provided question, For positive integers a and b,  we need to show that, xab=xab

Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

A:

Q: Show that if 9 | a3 + b3 + c3 then 3 divides at least one of the integers a, b, or c.

A:

Q: square of any odd integer is od.

A:

Q: В Calculate A and B so that + %D 1+x 1-

A:

Q: let a and b be real numbers. Prove by contraposition that, if a + b is greater than or equal to 100,…

A: Given that a and b be real numbers And a+b≥100 Then we have to prove that a>39 or b>60. Let…

Q: Let a be a positive real number. Evaluate va² – x?dx.

A: