MATH 312 Midterm 1
数学考试代考 2.(a) How many primes are of the form 15k + 5 for an integer k? Explain.(b) How many primes are of the form 15k + 6 for an integer k? Explain.
(a) How many primes are of the form 15k + 5 for an integer k? Explain.
(b) How many primes are of the form 15k + 6 for an integer k? Explain.
(c) How many primes are of the form 15k + 7 for an integer k? Explain.
Prove there exist infifinitely many primes using Euclid’s argument. You may assume for your proof that every integer greater than 1 has a prime divisor.
(a) Use the Prime Number Theorem to estimate π(40), the number of primes less than 40. Give your answer rounded offff to the nearest integer.
(b) Find the actual number of primes less than 40 by listing them all.
(a) Use the Euclidean algorithm to fifind the greatest common divisor of 2059 and 2581, then write the greatest common divisor as an integer linear combination of 2059 and Use one of the short methods.
(b) Use your result to fifind the least common multiple of 2059 and 2581.
(a) If a and b are integers such that a2∣b2 , prove that a∣b. [Hint: Fundamental Theorem of Arithmetic]
(b) Find integers a and b such that a2∣b3 , but a∣b.
In the homework you completed the proof of the following equivalence for positive integers d, e, f: (de, df) = d ⇔ (e, f) = 1. Use this fact, but NOT the Fundamental Theorem, to prove that if a, b, c are positive integers, then c(a, b) = (ca, cb). [Hint: Let d = (a, b).]
8.Find the integer m such that 2m∣ (70!).
9.Show that 257 is a factor of 2120 + 1 by using an algebraic identity.
For the following, state whether or not it is TRUE or FALSE. If it is true, provide justififi-cation or reference a result from class or the assignments. If it is false, you must provide a counterexample or explain why it is false.
(a) If an integer n divides ab for integers a and b, then it must divide one of the factors.i.e. n∣ab⇒ n∣a or n∣b for a, b, n∈ Z
(b) The greatest common divisor of two positive integers a and b can always be written as an integer linear combination of a and b
(c) The integers n and n + 1 are always relatively prime
(d) The integers n and n + 2 are always relatively prime
(e) For positive integers a and b, if a∣b then a2∣b
(f) For any prime p and positive integer n, if p‖n thenp‖n!
(g) If p is a prime, then p‖p!
(h) If 2m − 1 is prime, then m must be a power of 2