Let a and b be non relatively prime positive integers. If there exist integers s and t with as+bt=2 then' O gcd(a,b)=2 a and b are relatively prime O gcd(a,b)=4 O gcd(a,b)=3 None of these

Q: 5. Let n Your-Reg-ID (7 digits number) and = let 21, 22, 23,..., Zn be the (distinct) n-th roots of…

A:

Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

A:

Q: 2) How many positive integers of at most 3 digits are there such that the product of their digits is…

A:

Q: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

Q: Prove that the quadratic congruence 6x^(2) + 5x + 1≡  0 (mod p) has a solution for every prime p,…

A: We want to prove that the quadratic congruence 6x2+5x+1≡0 mod p has a solution for every prime p…

Q: e3x e(n+1)x = 1+ e" + 2 enx Sx 6. n! (n + 1)!

A:

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

Q: 6. Let a1, a2,...,an be distinct positive in not containing s = a₁ + a₂ + + an. A grassh A AL icht…

A:

Q: Find all natural numbers in such that n²<z". Prove your assertion.

A:

Q: 6. Let a be a number written (in base 10) as ao 10⁰+ a₁ 10¹ + a₂. 10²+...+an. 10" where 0 ≤a < 10.…

A:

Q: Let a and b be integers with GCD(a, b) = d and let a = da', b = db'. Prove that GCD(a', b')=1

A: and  be integer such that .It is given that  and .We have to prove .Note: According to the…

Q: Suppose that there exist integers x and y such that ax + by = 1. Show that a and b are relatively…

A:

Q: what

A:

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

A:

Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

A:

Q: 9(x + 6) - 41 = 75 Prove x - 62/9)

A:

Q: 5. Let (α1,α2,...,ân) and (b1,b2,...,bn) be two n-tuples 1 1 bers and let p, q be two positive real…

A: Provided that (a1,a2,a3,.....an)  and (b1,b2,.....bn) be two n-tuples of realnumbers and let p,q be…

Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

A:

Q: Let x be the set of all positive integers and d(min) = | = - = | show that (x,d) is not complete

A:

Q: Show that =- 5)" /31 + 6. 1 = 2 ) /7! + () /9! – ... %3D | -- IN

A: We know that, from the Taylor's formula for power series expansion of sinx is given as…

Q: If al = b (mod p), then prove that a = b (mod p).

A: (ii) given that                                                ap≡bp   (mod p) we have to prove…

Q: 11. What is the sum of all possible positive integers a, such that for any positive is an integer?…

A: Claim: n(n+2)(n+4) is always divisible by 3. Proof: we know, n = 0 or 1 or 2 (mod 3)    When, n = 0…

Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

A: .

Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

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

A:

Q: Prove that for all integers m>0 , m^2+5m+4 is composite.

A:

Q: 1.How do you find the polynomial if roots is given?

A: 1. We have to find the polynomial if roots is given.

Q: Show that /(-1)-7(1) but there is Let f(x) 1 Show that /(-1) (1) but there is no number e in (-1, 1)…

A:

Q: Let a = (1 73)(5 49 2) and B = (2 3)(7 4)(5 18) be elements of S10. Then, a B is O (12974)(3 58) O…

A: Let α=1735492 and β=2374518 be elements of S10, then α2β-1 is

Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

A:

Q: are 51 software students taking Let pi, p2,.. Psi be their final cam scores. Hence, cach p, is an…

A: Given that p1, p2, p3,..., p51 are the final exam scores of 51 software students who are taking MATH…

Q: 5. Identify the mistake(s) in the following "proof": Theorem: The sum of any two positive integers…

A:

Q: Let m 2q be an even positive integer. Prove that every integer is congruent modulo m to a unique…

A:

Q: Prae that x4 -5x +3 is ineducible ouer Q

A: Given polynomial is f(x)= x^4 -5x^3 +3

Q: a) For an odd prime p, prove that the quadratic residues of p are congruent mod p to the integers…

A:

Q: 10. Let m and n are two positive integers with m > n. Let a be a positive integer. Prove that a2" +1…

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.