2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution in Z regardless of the value of the integer b.

Q: (a) Solve the linear congruence 26x = 1 modulo 33

A: Solution

Q: Determine the positive real numbers x and y such that • x + y = 4 and 2.n2 + y² is small as…

A:

Q: Solve the quadratic congruence, 6x²-7x+2 ≡ 0 mod 31.

A: We have to solve the above quadratic congurence.

Q: Find the general solution to the system of congruence equations X ≡ 1 mod 6; X ≡ 4 mod 9; X ≡ 7 mod…

A: The given problem is to find the general solution for the given system of congruence equations and…

Q: 1. If x=27-3k and y = i +5j -4k, express each of the following in terms of i, j, k b) 2(3x+y)-3(4y…

A: Given x→=2i→-3k→ & y→=i→+5j→-4k→To write expression in terms of i→,j→,k→:

Q: 8. Let a, b and c be any positive real numbers. Prove that b² + c² a²+b² 2c a² + c² + 2b 2a > a+b+c.

A:

Q: aup 4. Find all solutions of the system of congruences 2x = 1 (mod 5), 3x4 (mod 7).

A:

Q: -8 -6 - 41 | + 3. 10 -6] Compute: 4 · 1

A:

Q: 5. Show that b, is always positive, by establishing the equation (Qn-1,Qn-1) (Qn-2,Qn-2) b,

A: We will apply the properties of inner-product to prove the given statement.

Q: (a+b)(a+ba)(ao³ + bw) = a' +b*

A: The question is wrong. Ask your guide for correction.

Q: Find the general solution to the system of congruences: x ≡ 3 (mod 7), x ≡ 4 (mod 13).

A: We have to find a general solution to the system of congruences:x ≡ 3 (mod 7),x ≡ 4 (mod 13).

Q: Which of the following linear congruence has 1 incongruent solution? (a) 3x = 9 (mod21) (b) 14x = 12…

A: The solution is given as

Q: Which of the following linear congruence has 2 incongruent solutions? (a) 3x = 9 (mod21) (b) 14x =…

A: to solve linear congurence , ax≡b(mod m) .........(1) the first stem is to reduce the congurence  by…

Q: LET Z Z₂ = FIND ZI 리 Z₂ 8+31 9-21 AND Z₂ Z₂

A:

Q: Prove that A2=I.

A:

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: 10) Prove the following statements true or false: i)āx(bx?) = (āxb)xã, ii)(a + b)x(ã − b) = 26 xã

A:

Q: Let A = 2 21 -1 1 1 0 3 3 ---B 3 Calculate A¹, and use A to solve Ax

A:

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: ndicate the least integer n for which (3x5 + x2)4 is O(xn).

A: Solved using the binomial theorem

Q: 6. Express in a + bi form: 2i(i + 3)

A:

Q: Exercise 8. [1, 2.2.10] Show that F₂Q1v₂ [c] iff a Vv₂Qcv₂ Here Q is a two-place predicate symbol…

A: Given information:A first-order logic structure denoted as A.Q is a two-argument predicate symbol.c…

Q: 1. If x=27-3k and y = 7+57-4k, express each of the following in terms of i, j, k a) 5x-2y

A:

Q: Solve the systems of linear congruence equations.

A:

Q: 3. Find all solutions of the system of congruences (mod 12), (mod 10). x = 5 x = 8

A:

Q: Show that if we make the change of variables =x – ct and n =x + ct, then, as in (16.7), a?u ,a²u a…

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Q: Suppose that |a| = 24. What is the generator of (a²l)n (a!o)? a20 a a14

A: Suppose that |a| = 24. We have to find the generator of <a21>∩<a10> .

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

A:

Q: Prove that {4a + 126 : a, b E Z} = {4c: c € Z}.

A:

Q: ind all solutions of the linear congruence: 9x = 5 mod (14)

A:

Q: Match the systems of linear congruences on the left to their solutions on the right. O Instructions…

A:

Q: Solve the linear congruence 7а + 3у %3D 10 (тod16).

A: Consider the linear congruence 7x + 3y ≡ 10 mod 16.Since gcd (7, 3) = 1 we know at least one…

Q: Let A = 1 3 0 a) A-BT -4 -3 8 5 and B= = b) -3B12+2A32 c) B₁* A*2 -2 3 -1 -1 4] -2 10 7 · Calculate…

A: a)A−BT=⎣⎢⎢⎢⎡​424−4​−7−1−2−2​⎦⎥⎥⎥⎤​ b)−3B12​+2A32​=19 c)B1∗​.A∗2​=23Explanation: