(b) Prove that multiplication mod n is commutative.

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: Let x be an integer. Prove that, if x mod 9 = 5, then 9|(x^2-2x+3)

A: Given: x mod 9=5 then, 9|x2-2x+3|

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

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: Use Fermat's theorem, show that if ged(a, 155) = 1 then a121 = a (mod 155).

A:

Q: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: Prove that if a and b are integers, then (a^5 + b^5) is congruent to a^5 + b^5 modulo 5.

A:

Q: For which positive integers a is the congruence ax4 ≡ 2 (mod 13) solvable?

A:

Q: 3. Show that ifn = 2 (mod 4), then 9" + 8" is divisible by 5.

A: Here we will prove the given statement.

Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: prove, for all integers x, if x mod 3 does not equal 0, then 3 | (x2 +2)

A:

Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

A:

Q: Establish that if a is an odd integer, then 2" aʻ = 1 (mod 2" *2) for any n 2 1.

A:

Q: Use Fermat's theorem, show that if gcd(a, 145) = 1 then a141 = a (mod 145).

A:

Q: Find the least positive integer n that satisfies the congruence 7 ^128 congruent to n (mod13)

A: Given, 7128≡nmod 13 Find the value of n that satisfies the above congruence.

Q: Let a, n be positive integers. Knowing that 15a - 13n = 1, we deduce that the multiplicative inverse…

A:

Q: 2

A:

Q: find the integer x such that 0 < x < 232 and 144x = 4(mod 233). Show your work.

A:

Q: Consider the congruence mod 6 for integers. Z6={[0], [1], [2], [3], [4], [5]}. Prove the product of…

A:

Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

A:

Q: If a mod b = 0, then b | a true or false

A: It's  pronounced “a is congruent to b mod n” (or “a is congruent to b modulo n”). Basically, it…

Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

A:

Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

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: 2. Find an inverse of 2 modulo 17 that is positive and less than 17.

A:

Q: Use Fermat's theorem, show that if ged(a, 155) 1 then a1 = a (mod 15

A:

Q: Prove directly from the definition of congruence modulo n that if a, c, and n are integers, n > 1,…

A:

Q: For every integer n &gt; 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

A:

Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

A:

Q: Suppose that a, b, m, and n are integers (n, m positive) and suppose we know a = b (mod n) and n | m…

A: We first show how the conclusion is true from that we choose the right option.

Q: Construct the addition and multiplication table for Z4 (Note that, both operations are mod4)

A: We have to solve given problem:

Q: Let 1Sasb be two positive integers. How can I prove or disprove that there exists a 1sks b-1 such…

A:

Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

A:

Q: a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the…

A:

Q: Give an example to show that a ^2 ≡ b^2 (mod m) need not imply that a≡ b (mod m)

A: Here I listed given example of above statement.

Q: Find a two digit integer c such that 101c = 1 mod 107. This congruence has a unique solution mod 101…

A: We need to find a digit integer c such that 101c ≡ 1 mod 107. As per the given question, this…

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

A:

Q: Prove by Induction that for every natural number п, 3h-1 is a multiple of 2?

A:

Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

A:

Q: Suppose a and b are relatively prime positive integers. Is it true that (a+2b)4(ab) = 1 (mod ab) ?…

A: We try to solve the question using Euler's Theorem. The detailed solution typeset in LaTeX is…

Q: If a (mod 0) = a, How would you interpret the congruence a ≡ b (mod 0)?

A: According to the given information, it is given that:

Q: Establish that if a is an odd integer, then 2" = 1 (mod 2" +2) for any n 2 1.

A:

Q: Prove that if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

A:

Q: Prove, for all integers x, if x mod 3 is not 0, then 3|(x2+2).

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let n be any fixed positive integer. Then…

Q: Prove that whenever x ∈Zand d ∈Z+, if d does not divide (x2 −3x), then x mod d does not divide 3

A: The given statement is False.  Consider, x = 5 and d = 4. Then we see, x2 - 3x = 52 - 3x5…

Q: Find all the integers that satisfy 140x = 133 (mod 301).

A:

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: Number theory

A: