3. Prove of disprove the following statements: (a) If a divides b and b divides a then a = b. (b) If l is the square of an odd integer, thenl =1 (mod 4).

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

A:

Q: Prove dis prove that the or smallest non- commutative ving is of order 4-

A:

Q: 14. Demonstrate that 21 has no primitive root.

A: An integer n≥2 has a primitive root if and only if the the number n is one of the following; 2,4,…

Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

Q: LI) Prove that (3

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

A:

Q: He have to Show that lon = (-1)" (mod 1") shall use mathematical moluction Let Pin) := lon a (47"…

A:

Q: prove that the following integer is composite (not prime). Do not use mod. 2^35(2^33-1) + 1

A:

Q: 5 is a quadratic non- residue mod 17 True False

A: 5 is a quadratic non- residue mod 17

Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n>=1 in question number (7) Sum of squares of first n natural numbers…

Q: 1. Prove that 5 is irrational.

A: Consider the provided expression, 5 Show that the above expression is irrational. We proof by…

Q: Prove that 2^10 + 5^12 is a composite number.

A:

Q: 8, Suppose a,b € Z. Prove that a = (mod 5). b (mod 10) if and only if a = b (mod 2) and a = b

A:

Q: 3. Use Wilson's Theorem to prove 6(k - 4)! = 1 (mod k:), if k is prime.

A: Wilson's Theorem: Let k be a prime number, then by Wilson's theorem, (k-1)!=-1(modk)

Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: 1. Prove that V3 is irrational number.

A:

Q: 4. Using your work from the earlier parts or independently find: (a) mod 8 (b) + mod 23 17 (c) mod 7

A: Given

Q: Prove by any method(contradiction/contrapositive or cases)   Prove that (a mod 2)(b mod 2)= (ab)…

A:

Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

Q: Prove the following statement directly from the definitions. The product of any two even integers is…

A: The product of any two even integers is a multiple of 4.

Q: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

A:

Q: Prove that for any odd prime p different from 3, ) = 1 if and only if p = ±1 (mod 12). (hint: Use…

A:

Q: : a number is even  if

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: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: 3. Show that (5+ 13)e + (5- i3)-" is a real number if is also a real number.

A:

Q: State whether True or False: If A and B are integers and A ≡ 3 (mod 3) and B ≡ 2 (mod 6) then A+B ≡…

A:

Q: of the apples is equal to 3 of the oranges. If there are 12 mor 5 In a fruit stall, - - annles than…

A: Fraction:  A small part of something.  In other words, numerical value that is not a whole number A…

Q: (3) prove That () 3

A: We have to use z = x + iy.

Q: . How many different values are there for 4" (mod 7), as n ranges over all positive integers?

A: Note - As per our guidelines we have to answer first question if multiple questions are posted. So…

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Prove that 21 divides 2n-) 4nt whenever is Posi tive in teger in

A:

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

A:

Q: Prove that no integer congruent to 3 modulo 4 can be written as the sum of two squares.

A:

Q: Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5).  Give 2 examples to…

A: Given, x,y∈ℤ and x≡1mod 5, y≡2mod 5

Q: Let's take an integer n to be odd if there is some integer k such that n = 2k – 1. Prove that the…

A:

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: 9. Use Euler's Theorem to show that 3P(26). =1 (mod 26). Show your work or explain your reasoning…

A: To show it by Euler's theorem :-

Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: 1. Prove that 3 is irrational number.

A:

Q: 3. Prove that 0 divides an integer a if and only if a 0.(In your proof. you can freely ILNE your…

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Q: Find the solution to 2x = 7(mod 17) that is positive and less than 17.

A: We find solution of given congruence relation

Q: What is the smallest positive integer x in the equarion: 1=3x+2(mod 7)?

A: We have to solve the equation 1 = 3x + 2 (mod 7)⇒1 - 2 - 3x = 0 (mod 7)⇒-1 - 3x = 0 (mod 7)⇒3x = - 1…

Q: prove that, it ņ is an 3n+2is oald then integer and nis add

A: