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Problem 5. (1 point) [5 Marks] Consider the following integer n = 516734400 = 2 ·...
Consider the following integer n= 4806841016250 = 2^1 * 3^6 * 5^4 * 7^4 * 13^3 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∈Z c) How many of the positive divisors of n are relatively prime with 6?
please post clear picture or solution. Bonus question: 4 bonus marks] A positive integer r is called powerful if for all prime numbers P, p implies p | r. A positive integer z is called a perfect power if there exist a prime number p and a natural number n such that p". An Achilles number is one that is powerful but is not a perfect power. For example, 72 is an Achilles number. Prove that if a and b...
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:...
Prove that there exists infinitely many numbers of the form an = n(n+1)/2 , for some positive integer n, such that every pair an, am (for n != m) are relatively prime. [Hint: Assume there exists a finite sequence an1 < an2 < an3 < . . . < anm, where nj are increasing positive integers. Show that using those numbers we can construct a new number that fulfills the requirements.]
1 1 point Consider the following algorithm for factoring an integer N provided as input (in binary): For i = 2 to [VN.17 i divides N, then output (i, N/). Which of the following statements is true? This algorithm is correct, but it runs in exponential time. This algorithm is not correct, because it will sometimes fail to find a factorization of Neven if N is composite This algorithm runs in sub-linear time, and always factors N it Nis composite...
(1 point) (3 Marks] Note that 0 € N. Give a recursive function f:N → N that represents the sequence ao, 21, 22, az... if an 10n + 1. (1 point) [3 Marks] Find mod(31004 + 1004!, 11). A. I am finished this question
(1 point) [3 Marks] Note that 0 € N. Give a recursive function f: N N that represents the sequence ao, 21, 22, 23... if an= 5n + 1. A. I am finished this question
Part 15A and 15B (15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
(1 point) [4 Marks] Consider the following statements Q : There exists a real number n such that n? > 100 implies n < 10 and n > 0 R: If Tom is Ann's father then Jim is her uncle or Sue is her aunt or Mary is her cousin In English, what are the negations, converses, and contrapositives of Q and R? You do not need to justify the correctness of the statements. A. I am finished this question
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use induction to prove that F. ends in digit 7 whenever n 2 is an integer (b) Use induction to prove that F= 2 + IT- Fholds for all neN. (c) Use (b) to prove that ged(F, F.) = 1 holds for all distinct nonnegative integers m, na (d) Use (e) to give a quick proof that there must be infinitely many primes! That is...