13. Let m, n, and k be positive integers. Prove that vp (mn) = vp(m) + vp(n) and vp(mk) = kvp (m).

Q: VnEN+, 5" + 3n > 22n+1

A: Given that the statement ∀n∈ℕ+ , 5n+3n≥22n+1. The objective is to prove the given statement.…

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: For positive integers m > n , show that -2 + n- 1 -3 + ...+ n + n-1 n - 1 + in

A: The right-side of the given equation can be expressed as…

Q: 5. Prove that, for n> 1, n

A:

Q: 1. Let P(A) = 0.3 and P(B) = 0.6. a) Find P(AUB) when A and B are independent. b) Find P(A/B) when A…

A: P(A) = 0.3P(B) = 0.6

Q: (2n- 2. Give a combinatorial argument to show that k- ways the number of ways to select a committee…

A:

Q: 8. Let dgcd(135, 351). Find integers m and n such that 135m + 351n = d.

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: 10. Find U 4, and N4,, where i is a positive integer if (a) A, = {i,i+1,i + 2,.) (b) A, = {0,i} %3D…

A: (a) Given that,  Ai=i, i+1, i+2,..... Then, A1=1, 2, 3,.....A2=2, 3, 4,.....⋮An=n, n+1, n+2,.....⋮…

Q: 3. Is this sentence consistent? “There is someone who loves exactly those people who do not love…

A:

Q: E. Show that both R and R² may be covered by countably many open balls.

A:

Q: Then, there exists k1≠0, k2≠0  such that a,b=k1c,k1d   and  c,d=k2e,k2f. a,b=k1c,d=k1k2e,…

A:

Q: 1) What is the sum of 8y 1- 2y? and -3y2 + 2y + 7?

A:

Q: 11. Prove that if d divides a, then dk divides ak for every positive integer k.

A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

Q: Expand la In 3 Inx Iny? Inz 2n 3+ 2mx+ 2lmy hz a. 2h3+ 2hx+ 2ny- 2hz In 3+ Inx+ 2hy- nz

A:

Q: 9. Determine the value(s) of k such that (2k, 3, k + 1) x (k-1, k, 1) = (1, -2,2).

A: (2k, 3,k+1)*(k-1, k, 1)=(1,-2,2)Find the value of k.

Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: Suppose that P(A) = 0.1, P(B) = 0.3, and P(A|B) + P(B|A) = 0.8. Find P(A ∩ B).

A: From the information, given thatThe formula of the conditional probability, for any two events…

Q: Suppose 50 percent of the customers at Pizza Palooza order a square pizza, 72 percent order a soft…

A: Given information- Percentage of the customers order a square pizza = 0.50 Percentage of customers…

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: 10.

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Q: Hence show that for positive integers n and m: P = I P P² for n = 3m for n = = 3m + 1 for n = 3m +2

A:

Q: 4. Let A

A: Given,                A=1, 2, a, b, ϕ

Q: The number of the element(s) of order 17 in Z17 O Z17 288 16 17 None of them 289

A:

Q: 31 Let A =-1 4 and B = 3] 2 [3 -1 . Compute the (3, 1) entry of AB.

A: Given: A=23−1403  and  B=3−13124[AB](3,2)=?[AB](2,3)=?

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

A:

Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

A:

Q: «А B 1)

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Q: Suppose J3 is derived from switching the rows of 13. -5 3 0 -8-1 7 -4 If AJ3 = 0 -2 7 -2 3 -5 -8 or…

A: NOTE: Refresh your page if you can't see any equations. . the 3×3 identity matrix is given by now…

Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: Suppose that P ( A ) = 0.4,  P ( B ) = 0.5 and A , B are disjoint. Compute P ( A c ∪ B c ) .

A: From the provided information, P (A) = 0.4, P (B) = 0.5 The events A and B are disjoint.

Q: 7. Establish for A is m × n and B, C are n x p, that A(B+C) = AB + AC

A: