If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b3 c (тod n) (b) b2 = c2(mod n) (c) ab = ac(mod n) (d) All the above.

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: TRUE or FALSE: 1. If a is congruent to b (mod m), then a^k is congruent to bk (mod m) for every…

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Q: Suppose b is any integer. If b mod 12 = 5, what is 6b mod 12? In other words, if division of b by 12…

A: Given b is an integer such that b mod12=5 To find the value of  6b mod12

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: What are the FOUR values of x that will satisfy: x ≡ 4 mod 5 & x ≡ 3 mod 7 please show your…

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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)…

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

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Q: True or False : a – c ≡ b – c (mod n) ⟺ a ≡ b (mod n)

A: We have to solve given problem:

Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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

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

A: Here we will prove the given statement.

Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: If m = 2* is a power of 2, explain how you could use repeated squaring to compute am (mod n) for all…

A: a) We have m = 2k. First we illustrate the method of repeated squaring to compute am (mod n). Then…

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

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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: Suppose b is any integer. If  b mod 12 = 5,  what is 6b mod 12? In other words, if division of b by…

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Q: Let P be a prime, if ac = bc (mod p) and p does not divide c, then a = b (mod p) Select one: O True…

A: Given  gcd(c, p) = 1 and ac ≡ bc(mod p). (We may assume p > 1 so that c  ≠  0.) Then ac − bc =…

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

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

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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: you Prove the " of 25nY-18n +7 =0 (mod4), then statement contradiction. is by "even

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Q: () 2p n! (a) (") = 2 (mod p), where k for 0 <k <n. k!(n – k)! (b) (Зр - 1)! 2р2 (mod p3)

A: As per our guidelines we are supposed to answer only fast question. Kindly repost other part in the…

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: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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Q: How do I prove this: Show that if x ≡ 1(mod n), then x^2 ≡ 1(mod n) and x^3 ≡ 1(mod n) and how to I…

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Q: 122 is congruent to 5 modulo17 ?

A: To check whether -122 is congruent to 5 modulo 17 or not

Q: The last two digits of 52044 is 44. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). F 10. F…

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Q: The inverse of 3 modulo 7is -2

A: Given statement, The inverse of 3 modulo 7 is -2.

Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

Q: Which of the following are true modulo 12? (1) Does 5z = 6 mod 12 have a solution? How many? (2)…

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Q: The remainder when aa is divided by 77 is 22 and the remainder when bb is divided by 77 is 55. (So a…

A: Given:          a≡2 mod(7)  and b≡5 mod(7)

Q: I know that using a corollary that the congruence x2 ≡ 0 (mod p) that the only solutions are + and -…

A: As per the question we have the following congruence relation : n ≡ 2(mod 4) And we have to prove…

Q: Consider the equation 2x + 1 = 0 mod n. For which n are there no solutions? one solution? multiple…

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Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: Inverse of 9 modulo 23 is 18

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Q: 23. Let a,be Z and n eN. If a =b (mod n), then a = ab (mod n). (24, If a b (mod n) and c=d (mod n),…

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Q: Suppose that 3x ≡ 2 (mod 5). Find the remainder when x is divided by 5; in other words, compute x…

A: ANSWER :

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

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Q: 10. Use Euler's Theorem to show that 5P(16), '=1 (mod 16). Show your work or explain your reasoning…

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Q: A sequence of pseudorandom numbers is generated as follows Xo = 4 x₁ = ( 6x₁-1 + 5 ) mod 13 if i > 0…

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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: Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not…

A: Given that 2n-12≡kmodn Take the square on both sides, 2n-12≡kmodn2n-12·2≡k2modn2n-1≡k2modn (a) The…

Q: Compute 22 mod 3, 24 mod 5, 26 mod 7.  What do you notice?  Does Fermat predict your answers? (If…

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Q: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

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Q: 20. If a e Z and a = 1 (mod 5), then a² = 1 (mod 5). If a - h (mod n) then a3 = h3 (mod n)

A: Notation: a|b  means a divides b or                   a|b means b is divisible by a.

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

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