L-) Find all in tegers such thet 5x = 12 (mod 7) 10x = 4 (mod 11) E 16 (mod 10d ond 1occ < x < 3000

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: 2-) Find all in tegers x such thet 5x = 12 (mod 7) 10x = 4 (mod 1) 7x E 16 (mod 10d ond 1occ < x <…

A: Given: 5x≡12(mod 7) 10x≡4(mod11) 7x≡16(mod 10) and 1000≤x≤3000

Q: x^2 ≡ 1 (mod 105)

A: Given: x2≡1(mod 105) To Solve: The given congruence  x2≡1(mod 105)

Q: I need help with this problem and an explanation for the solution described below (Discrete…

A: The objective of the question is to find the negation of the given statement. In logic, the negation…

Q: which member of binomial expression 10 6, of

A: Binomial expression=2x3+x−110According to Binomial theorem, a+bn=∑k=0nnkan−kbk

Q: Kample tetx be Let X€2Z. Prove that for cveri integer n2 , x^is cuen × is aven 2.

A: Yes, for n greater than or equal 2(n is a positive integer) if x is even then x^n will also be even.…

Q: For each statement, indicate whether it is true or false. Then prove or disprove it. (a) For all SC…

A: (a) FALSE.  Consider S ⊆ℤ, defined by  S = { 2+8k | k ∈ℤ } U { 2+12r | r∈ℤ }.  Then the given…

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 a,n∈N. Assume that n≥2. Show that if a^(1/n) is rational, then it is an integer.

A: The detailed solution is as follows below:

Q: [a, b] = N=1 (a -,b+-) To prove this, show the following: То Show that [a, b](a – ÷,b +- ) for every…

A: Given that, [a,b]=∩n=1∞a-1n,b+1n This can be written as, ∩n=1∞a-1n,b+1n=[a-1,b] The aim is to show…

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

A:

Q: (7) (Z/50Z)*. Which of the other elements of (Z/50Z)* are also primitve roots modulo 50? Find a…

A:

Q: a be an odd integer. Pro

A:

Q: Let p be a prime and a a natural number relatively prime to p. Let k be the smallest natural number…

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

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

Q: Find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime. Justify that your…

A: We have to find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime.

Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: If a≡3 (mod 11) and b≡9 (mod 11): (a) Compute a+b(mod 11) (b) Compute a^3(mod 11)

A:

Q: solve for x^2 = 1 mod 8 and x^2 = 1 mod 10

A: Solution...

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

A:

Q: Find the proportion of primes p in the interval from 10^12 and 10^12 + 1000 such that p-1 is B= 105…

A:

Q: List all numbers x such that 0≤x<100 and x ≡ [3,1] mod [4,5]

A: Given,           0≤x&lt;100, and  x ≡ [3,1] mod [4,5].

Q: show that 2 is a primitive root mod 3^e for all e>=1

A:

Q: Least residue of 45^37 mod 19 when using Fermats Little Theorem

A:

Q: Use Euler's Theorem to find a number x, 0 <x<7, such that x^21 = 3 (mod 8)

A: By Euler's theorem, for any number x such that 0 &lt; x ≤ 7,        xφ(8) ≡ 1 (mod 8)       =&gt; x4…

Q: Compute (7^3)^161 (mod 250)

A:

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

Q: n —(2) - (-) (1)¹(1)(). (n-1)

A:

Q: is the following fucntion analytic in mod(z)&lt;2 and/or mod(z)&gt;2. sqrt(z^(2)+2z+2)/ z^(2)-4z+3

A: The objective of the question is to determine whether the given function is analytic in the regions…

Q: 10. By Binomial Theorem (2a – b)* =16a* – 32a’b+24a*b? – 8ab³ +b* | a. TRUE. b. FALSE.

A: We know the binomial therom expansion

Q: Find a such that 5= 7* modulo 13.

A: Given: 5≡7xmodulo13. The objective is to find the value of x.

Q: Let A = {4n^2 + 1 | n ∈ Z}. Show that the size of A is equal to the size of Z≥0. Z≥0 is the set of…

A: A set A is said to be countable  if there is a bijective map between A and set of natural numbers .…

Q: Find the smallest positive integer k satisfying the stated condition or explain no such k exists:…

A: To find the smallest positive integer k satisfying the given condition of congruence.

Q: let n be a positive integer not exceeding 20. How many values of n such that the equation x^2 +…

A: We have to find the values of n for which x^2+y^3n+1=1.

Q: Let p=17. If a belongs to F_{17} and a is non zero , compute the value of a^{16}(mod 17) for a=3 .

A:

Q: Fermat.

A:

Q: 6. Let f(x) be a polynomial with integral coefficients (like f(x) = 3x^^5-x^2+7) and suppose there…

A: Given:- fx=3x5-x2+7 There are 2 solutions  x mod 5 such that fx is congruent to 3 mod 5 and there…

Q: Find all solutions of quadratic congruence x^2+85x+2=0(mod401)

A: We follow the method of completing the squares of the equation x^2+85x+2. Note that 85 is odd so we…

Statements and Logic

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