IfOcy (mod n) and e> 0 then a) *= y (mod cn) b) x= y (mod n) ) x=y (mod) d) None of these :Select one d. C a b

Q: U(12) under multipilcation modulo 12

A: To find U(12) under multiplication modulo 12.

Q: Za) Let A = {n E Z | n = 0 (mod 2)} and B = {n € Z | n = 0 (mod 4)}. Prove that if n E (A – B), then…

A: Follow the steps below.

Q: Prove that if a = b (modn) and m/n, then a = b (modm).

A:

Q: if b is modulo m and a is another unit with a=b mod m, we say that d is the---------of a modulo m to…

A: As with such a large amount of concepts we are going to see, congruence is straightforward, perhaps…

Q: Assume that k mod 15 4. Compute 11k mod 15. Show all your work clearly.

A: Topic:- basic maths

Q: 2. Give an example to show that a? = b2 (mod n) need not imply that a =b (mod n). 3. If

A: Given below the detailed solution

Q: If gcd(a, n) = 1, then the congruence ax = b modn has the solution x = - ba®(n)-1

A: Given that gcd(a,n)=1. We need to check whether ax≡b mod n has the solution x=ba∅(n)-1

Q: consider the statement: for all n∈N , x∈Z , x^2 ≡ 1 mod n (        ) x ≡ 1 mod n or x≡ -1 mod n a)…

A:

Q: Prove that if ax = b (modn) then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x =…

A:

Q: Show that 3n^ 2+8nlogn is big-Oh (n ^2 )

A:

Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

A:

Q: Compute the addition modulo operation 3+2 2 0. 1 3 O O

A:

Q: 5- If a = b mod(n), then ac = bc mod(n). 6- If a b mod (n), then ak = b*mod(n) for every positive…

A:

Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

A:

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

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: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: a) Show that ac = bc mod cd if and only if a = b mod d

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: Number theory

A:

Q: 2. In Section 3.1, we defined congruence modulon where n is a natural number. If a and b are…

A:

Q: 2. Give an example to show that a (mod n). b (mod n) need not imply that a =b

A:

Q: 1. Show that an integer n is divisible by 2 if and only if its last digit is even. (Hint: For each i…

A:

Q: 5. Explain why the congruence r = 2 (mod 4) has no solutions in Z.

A: We have to show that the congruence x2≡2(mod 4) has no solution in ℤ

Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: What is 2 ^32 mod (5)?

A: mod is the function whose output is always positive. mod is represented by .

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

A:

Q: 6), then the congruence has a unique solution modulo p. 11. Show that the congruence x³ = 3 (mod 19)…

A: 11. To prove that the congruence x3≡3 mod 19 has no solutions, whereas x2≡11(mod 19) has three…

Q: 11. Define conguren ce class es for modulo 4.

A:

Q: Explain why the congruence r2 = 2 (mod 4) has no solutions in Z

A: Result: Square of all integers modulo 4 is either 0 or 1.…

Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

Q: If a+c=b+c (mod m) then a =b (mod m) Select one: True False

A:

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: 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: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

A: I have proved it by using defination of congruent.

Q: 3) If the gcd(z,n) = d and xz = yz mod n then: Show that x = y mod

A:

Q: 25 Prove that mod 7 5 for all Gindegars and bin b mod 7-6

A:

Q: Solve the system of congruence's x = 2 (mod 3), x = 4 (mod 5), x = 1 (mod 7)

A:

Q: Show that : If n is an integer then n is congruent to 0 modulo 4 or 1 modulo 4.

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: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False

A:

Q: Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence…

A:

Q: If a # 0 (mod m) and b # 0 (mod m), then ab# 0 (mod m). Select one: O True False

A: justification is given in step 2)

Q: Suppose ab≡0 mod n and gcd(a, n) = 1. Apply Euclid’s Lemma to prove b≡0 mod n.

A: Given: ab≡0mod n and gcd(a,n)=1

Q: (a) gcd(q, n) = 1 if and only if kq # 0 mod (n) for all 1 < k< n. %3D

A:

Q: (2) If a = b modn and a = b mod m and ged(m, n): 1. then a =b mod mn.

A: The greatest common divisor of two or more integers, which are not all zero, is the largest positive…

Q: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False

A: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False

Q: Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p

A: