The classic equivalence classes are the ones that deal with modular arithmetic.  As you look at these examples, remember that numbers that are equivalent mod 4 for example, are related based on the remainder we get when we divide by 4. So 1, 5, 9, 13, … are all equivalent mod 4. What is common about all of these? Can you develop a general expression for any number that is equivalent to 1 mod 4?  How do you know your expression is correct? What if you did the same for the values that are congruent to 2 mod 4? What about 3? What do you notice?

Q: 1. In this problem we are working over the field Z7, the integers mod 7. All arithmetic is done mod…

A: In this problem, we are working over the field Z7, the integers mod 7. We use all arithmetic is done…

Q: The goal of this exercise is to practice finding the inverse modulo m of some (relatively prime)…

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Q: How many integers from 1000 through 9999 must you pick in order to be sure that at least two of them…

A: As per our guidelines we are supposed to answer only? One question ( if there are multiple questions…

Q: The modular representation of the integer 23 with respect to the modulo 2, 3, 5 is (1,2,3) (3,2,1)…

A: the modular representation of the integer 23 with respect to the modulo 2,3,5 is it is known that…

Q: How many integers from 1000 through 9999 must you pick in order to be sure that there is at least…

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Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡ −1 mod 7.

A: Since you have asked multiple questions, we will solve the Question Q2 for you. If youwant any…

Q: List five different natural numbers that are congruent to 8 modulo 6.

A: # as per the guidelines we are entitled to solve one question at a time ,please resubmit the other…

Q: Books are identified by an International Standard Book Number (ISBN), a 10-digit code x1x2... x10,…

A: Let us start finding q and m values

Q: For some of the most interesting laws of modular arithmetic (Choose some of your liking, but do…

A: This is a problem of mudular arithmatic.

Q: When performing modular arithmetic using modulo 3, all the integers are expressed in only three…

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Q: The Modular Operation r mod m = r denotes that r is the remainder = 3. If two integers have the same…

A: Rewrite the following theorems using quantifiers and predicates.

Q: Two great mathematicians who have contributed greatly in the field of modular arithmetic are Swiss…

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Q: Complete the following multiplication table for mod 9 AND use it to identify all the congruence…

A: Solution: The multiplication table modulo 9 is x 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 1…

Q: Show that every nonzero residue (mod 17) can be written as a power of 3

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Q: The set {0,5,6} is closed under Mod(10)

A: To check if { 0 5, 6} is closed under mod(10)

Q: How many integers have inverses modulo 144? Justify.

A: Justify how many integers have inverse modulo 144.

Q: Find the inverse s of -1959 modulo 979 such that 0 <s< 979

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Q: Hello, Please help me write this in either a direct or contrapositive proof (whahtever is the most…

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Q: What prime numbers are squares in mod 17? (Include primes that are larger than 17). What primes p…

A: Since the squares less than 17 are 1, 4, 9, 16 then the prime numbers that are squares in mod 17 are…

Q: If you multiply n whole numbers (even or odd) when is the answer odd? Translate into multiplication…

A: If we multiply whole numbers, the pattern is as follows: Even × Even=Even Even × Odd=Even Odd ×…

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

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Q: #5) Same as in #4 but now regarding (37603|48611) as a Jacobi symbol and using the Law of Quadratic…

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Q: (2) Use the power property of modular arithmetic and a technique discussed in section 8.4 to…

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Q: 8. Use Euler's Theorem to find a number a between 0 and 9 such that a is congruent to 7 1000 modulo…

A: We know according to  Euler's Theorem  that  xϕ(n)≡1(mod n) where the function ϕ(n) counts the…

Q: If a = 17 (mod 3), then what is the remainder of a when divided by 3? Enter the numerical value.

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Q: The multiplication rule says that if an operation can be performed in k steps and, for each i with 1…

A: Answer is mentioned below

Q: Use Fermat’s little theorem to evaluate 9200 mod 19.

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Q: 1.21 Theorem. Let a natural number n be expressed in base 10 as  n=akak-1...a1a0 (Note that what we…

A: We know that 10 mod 3 = 1. So,

Q: 10. Use Euler's Theorem to show that 516) = 1 (mod 16).

A: Euler's theorem states that          "If gcd(a, n) = 1, then aφ(n) = 1 (mod n)

Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

Q: whether 2 is a primitive root modulo 19. Please state the reason.

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Q: Books are identified by an International Standard Book Number (ISBN), a 10-digit code x1x2... x10,…

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Q: When we work with a congruence class, we have to keep two key things in mind. First, we have to know…

A: For ℤ3, The ones that are congruent to 0 is the set {...,-6,-3,0,3,6,..}. We call it the congruence…

Q: A safer system would be to use modular multiplication instead of addition. But if we use modular…

A: The modular arithmetics gives remainder obtained upon division. For example, 26≡1mod 25 means on…

Q: 3. Find the last two digits of the number 99°. [Hint: 9° = 9 (mod 10); hence, 99 = = 99+10k. notice…

A: In question number 3: we have to find the last two digits of the number (((9)^9)^9) That means to…

Q: Find a number that is less than 1740719 which is congruent to 577 modulo 1279 and congruent to 317…

A: Given, a number that is less than 1740719 which is congruent to 577 modulo 1279 and congruent to 317…

Q: Use Chinese theorem to find all positive integers that makes this congruence true:        x = 2 mod…

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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: 1. In this problem we are working over the field Z, the integers mod 7. All arithmetic is done mod…

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Q: (i) 31000 mod 13

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Q: When performing modular arithmetic using modulo 3, all the integers are expressed in only three…

A: We need to find the numbers in the residue classes modulo n.

Q: Use modular arithmetic to determine the following:  Disregarding a.m or p.m., if it is now 9…

A: If a.m. or p.m. is disregarded, and the time is x o'clock now, then: The time, y hours from now,…

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: 1. Decimal System – base 10; 0,1,2,3,4,5,6,7,8,9 2. Binary System - base 2 3. Octal System - base 8…

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Q: The following is a list of converse statement of modular arithmetic. Determine if any of the…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: How would you find -88 mod 13?

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Statements and Logic

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