Let k and a be two positive integers, such that ak-1 = 1(mod k) and gcd(k, a) = 1. Is k prime or composite? If so why and explain all the steps. Thanks
Let k and a be two positive integers, such that ak-1 = 1(mod k) and gcd(k, a) = 1. Is k prime or composite? If so why and explain all the steps. Thanks
For any two positive integers a, b, define k(a,b) to be the largest k such that a* | b but ak+1b. Given two positive integers x, y, show that (a) k(a, gcd(x, y)) = min{k(a, x), k(a, y)} for any positive integer a (b) k(a, lcm(z, y)) = max{k(a,a),k(a, y)} for any positive integer a. Hint: Think of the prime factorization of the numbers
For any two positive integers a, b, define k(a,b) to be the largest k such that...
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...
Problem 1. Let a be any positive integer relatively prime to 10 (so gcd(a, 10) 1). Show there are an infinite number of multiples of a whose decimal expansion is all 9s. For example, 13 we have 13 · 76923 = 999999 and 13 · 76923076923 = 999999999999. (Hint: a number whose decimal expansion is all 9s is of the form 109 – 1). Do you notice any connection here to the decimal expansion of the fraction 1/a? for a...
Let p be a prime. If for integers k and I we have rk = x (mod p) for all x E Z, (x,p) = 1 show that k =l (mod p – 1).
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.
How many integers k with 1 S k < 550 have GCD(k, 550)- 1? If n 〉 2 is an integer with o(n) -1, explain why n must be prime
How many integers k with 1 S k
PLEASE SHOW ALL STEPS WITH EXPLAINATION Let m and n be positive integers and let k be the least common multiple of m and n. Show that mZ∩nZ=kZ.
1. (10 points) GCD Algorithm The greatest common divisor of two integers a and b where a 2 b is equal to the greatest common divisor of b and (a mod b). Write a program that implements this algorithm to find the GCD of two integers. Assume that both integers are positive. Follow this algorithm: 1. Call the two integers large and small. 2. If small is equal to 0: stop: large is the GCD. 3. Else, divide large by...
Let m be a positive integer and let a and b be integers relatively prime to m with (ord m a , ord m b) )=1. Prove that ord m (ab)= (ord m a) (ord m b) (Hint: Let k=ord m(a),l=ord m(b), and n=ord m(ab). Then 1≡(ab)^kn≡b^kn mod m. What does this imply about l in relation to kn?
PYTHON In mathematics, the Greatest Common Divisor (GCD) of two integers is the largest positive integer that divides the two numbers without a remainder. For example, the GCD of 8 and 12 is 4. Steps to calculate the GCD of two positive integers a,b using the Binary method is given below: Input: a, b integers If a<=0 or b<=0, then Return 0 Else, d = 0 while a and b are both even do a = a/2 b = b/2...