How many integers from 1000 through 9999 must you pick in order to be sure that at least two of them have the same remainder modulo 79? Explain your reasoning.

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: Verify the following facts about Fibonacci number: If we make a list of any four consecutive…

A: Here the two different four number list of Fibonacci  number with the property.

Q: The number of factors of prime numbers are None of the mentioned O 30 Depends on the prime number

A: No. Of factor of prime no. Means how many divisor consist of a prime no.

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: 2.3 How many integers between 0 and 500: 1. Have distinct digits? 2. Are divisible by 3? 3. Are…

A: Formulae used: Total number of elements in the events E1 or E2 : n(E1∪E2)=n(E1)+n(E2)-n(E1∩E2) Total…

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

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Q: 12. Prove that there are infinitely many prime numbers in the arith- metic progression 1,4,7, 10,…

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Q: modulo 5   B).  Evaluate 31mod5

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Q: The order of the integer 4 modulo 19 is: O 4 O None of the choices O 6

A: To find The order of the integer 4 modulo 19.

Q: How many numbers between 1 and 10000 have the digits 1,3, and 5 such that each digit appears exactly…

A: Here Numbers between 1 and 10,000 have digits 1,3,5 and each appears exactly once only

Q: The classic equivalence classes are the ones that deal with modular arithmetic.  As you look at…

A: Consider the equivalence class of integers with respect to the relation "mod 4" i.e the remainder…

Q: (ii) The first few cubes are 1,8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, and 1728. Write 1729…

A: The number 1729 is known as a Carmichael number. It can be represented as the sum of two cubes. For…

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: If you randomly pick integers from 1 to 200, how many would you have to pick to ensure that at least…

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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: What is the second step in proving by mathematical induction that for every positive integer n, 11"…

A: We have to provide the second step of mathematical induction for every positive integer n, 11n-6…

Q: 5. How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

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Q: If n is prime, then 2" – 1 is prime.

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Q: The order of the integer 8 modulo 13 is: O 6 O 4 O None of the choices O 2

A: To find The order of the integer 8 modulo 13.

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: 1)How many integers between 1000 and 2000 are divisible by at least one of the integers 3, 5 and 7?

A: Introduction: In mathematics, integers are the sum of positive and negative numbers. Integers are…

Q: 2. Create the following tables: (a) Addition table for modulo 7 (b) Multiplication table for modulo…

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Q: Determine how many ti (o bom is a prime.

A: In this question, the number of times is determined when 1 appears in the multiplication table for…

Q: Suppose you are using the fast method to compute xe mod N, where x=23, e=101 and N=515. You have…

A: We will find out the required value.

Q: Find the order of 4 modulo 49.

A: Evaluate order of 4 modulo 49.

Q: 3. The Lucas Lehmer Test is applied below to test whether the number 255 is prime or not. (S,' -2 k…

A: The Lucas Lehmer test: Let Mp =2p-1  be the Mersenne number to test where p is an odd prime.…

Q: Let c = 2^100 and p be the smallest prime such that p >= c. How many composite numbers are between c…

A: Let c=2100 And p=min p is prime:p≥c,p∈ℤ Claim: Total number of composite number between c & p,…

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

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Q: Determine the remainder of 102020 modulo 2020 and the remainder of 102020 modulo 2021.

A: solution 104≡1920(mod2020)…

Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

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

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: 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: 29. How many different values are there for 4n(mod 7), as n ranges over all positive integers? 30.…

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Q: Which of the following is not a possible value for y ≡≡ 17 mod 8 in the equation ? Group of answer…

A: Consider the given equation.

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: Two integers are congruent modulo to a)prime integer b)odd integer C) not a prime nor composite…

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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: 3. If a = b (mod n), prove that gcd(a, n) = gcd(b, n). 4. (a) Find the remainders when 250 and 4165…

A: Given below the detailed solution

Q: What is the largest n such that 16 and 25 are in the same remainder class modulo n? Write out the…

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Q: 5. Find the last digit of 6. Find the last digit of the decimal expansion of 799.9. noritive residue…

A: We need to find the last digit of the decimal expansion of 7999,999.

Q: True or False Let n > 0 be fixed and a, b, c and d are arbitrary integers, then, if a ≡ b(mod n)…

A: We have to check whether the statement, "Let n>0 be fixed and a, b, c and d are arbitrary…

Q: Find the order of 5 modulo 31.

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Q: 10) Use the primitive root g = 3 modulo 31 to find a square root of 5 modulo 31. 11) Use the…

A: We will solve both these questions for you. Note that using these questions requires some basic…

Q: (a) Let n be a positive integer. Prove that n is congruent to the sum of its digits modulo 9. Hint:…

A: (a) First we will prove that 9(10m-1) for every m∈ℕ : Since, 10≡1mod 9 , we can write this as  910-1…

Q: 2. Suppose that instead of proving the claim "there is no integer n such that n= 2 mod 6 and n7 mod…

A: consider an integer n=26 such that n mod 6=26 mod 6               =2 mod 6 also, n mod 9=26 mod 9…