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Bonus question: 4 bonus marks] A positive integer r is called powerful if for all prime numbers P...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
8. (a) Prove that if p and q are prime numbers then p2 + pq is...
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b...
please prove proofs and do
7.4
7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
Recall that an integer >1 is called a prime when its only strictly positive factors are 1 and r. An integer > 1 is called composite when it's not a primec. (a) Show that a composite integer...
Recall that an integer >1 is called a prime when its only strictly positive factors are 1 and r. An integer > 1 is called composite when it's not a primec. (a) Show that a composite integer 2 < x < 150 must be a multiple of 2, 3, 5, 7, or 11 (b) Use the Sieve Method and a table with 15 rows and 10 columns to determine all primes between 2 and 150. (c) What's the largest prime...
Please prove the 3 theorems, thank you! 7.6 Theorem. Let p be a prime. Then half the numbers not congruent to 0 modulo p...
Please prove the 3 theorems,
thank you!
7.6 Theorem. Let p be a prime. Then half the numbers not congruent to 0 modulo p in any complete nesidue system modulo p are quadratic residuess modulo p and half are quadratic non-residues modulo p. From clementary school days, we have known that the product of a pos- itive number and a positive number is positive, a positive times a negative is negative, and the product of two negative numbers is positive....
10. Two integers r, y are called coprime if the only positive number that divides both...
10. Two integers r, y are called coprime if the only positive number that divides both of them is 1. Provide an infinite sequence of distinct natural numbers 11, 12, 13, ... such that none of these numbers is a prime number, yet any pair of them are coprime with each other. Justify your answer. [5 marks]
Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r)...
Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r) = p. Then r is irreducible in Z[i]. In particular, if a and b are integers such that a² +62 = p, then the Gaussian integers Ea – bi and £b£ai are irreducible. Exercise 16.1. Prove Theorem 16.1. (Hint: For the first part, suppose st is a factorization of r. You must show that this factorization is trivial. Apply the norm to obtain p=...
Problem 5. (1 point) [5 Marks] Consider the following integer n = 516734400 = 2 ·...
Problem 5. (1 point) [5 Marks] Consider the following integer n = 516734400 = 2 · 37 · 52 · 72 · 133. a) How many positive divisors does n have? b) How many of the positive divisors of n are perfect cubes? That is, the number can be written as (k)3 for some k e Z. c) How many of the positive divisors of n are relatively prime with 21? A. I am finished this question
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
*** Write a function called circular_primes that finds the number of circular prime numbers smaller than...
*** Write a function called circular_primes that finds the number of circular prime numbers smaller than n, where n is a positive integer scalar input argument. For example, the number, 197, is a circular prime because all rotations of its digits: 197, 971, and 719, are themselves prime. For instance, there are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97. It is important to emphasize that rotation means circular...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
please prove proofs and do
7.4
7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
Recall that an integer >1 is called a prime when its only strictly positive factors are 1 and r. An integer > 1 is called composite when it's not a primec. (a) Show that a composite integer 2 < x < 150 must be a multiple of 2, 3, 5, 7, or 11 (b) Use the Sieve Method and a table with 15 rows and 10 columns to determine all primes between 2 and 150. (c) What's the largest prime...
Please prove the 3 theorems,
thank you!
7.6 Theorem. Let p be a prime. Then half the numbers not congruent to 0 modulo p in any complete nesidue system modulo p are quadratic residuess modulo p and half are quadratic non-residues modulo p. From clementary school days, we have known that the product of a pos- itive number and a positive number is positive, a positive times a negative is negative, and the product of two negative numbers is positive....
10. Two integers r, y are called coprime if the only positive number that divides both of them is 1. Provide an infinite sequence of distinct natural numbers 11, 12, 13, ... such that none of these numbers is a prime number, yet any pair of them are coprime with each other. Justify your answer. [5 marks]
Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r) = p. Then r is irreducible in Z[i]. In particular, if a and b are integers such that a² +62 = p, then the Gaussian integers Ea – bi and £b£ai are irreducible. Exercise 16.1. Prove Theorem 16.1. (Hint: For the first part, suppose st is a factorization of r. You must show that this factorization is trivial. Apply the norm to obtain p=...
Problem 5. (1 point) [5 Marks] Consider the following integer n = 516734400 = 2 · 37 · 52 · 72 · 133. a) How many positive divisors does n have? b) How many of the positive divisors of n are perfect cubes? That is, the number can be written as (k)3 for some k e Z. c) How many of the positive divisors of n are relatively prime with 21? A. I am finished this question
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
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