Consider the following integer n= 4806841016250 = 2^1 * 3^6 * 5^4 * 7^4 * 13^3...
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?
Homework Answers
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Consider the following
integer
n= 4806841016250 = 2^1 * 3^6 * 5^4 * 7^4 * 13^3...
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
Challenge activity: A partition of a positive integer n is the expression of n as the sum of positive integers, whe...
Challenge activity: A partition of a positive integer n is the expression of n as the sum of positive integers, where order does not matter. For example, two partitions of 7 are 7 1+1+1+4 and 7=1+1+1+2+2. A partition of n is perfect if every integer from 1 to n can be written uniquely as the sum of elements in the partition. 1+1+1+4 is perfect since 1-7 are expressed only as 1, 1+1, 1+1+1, 4, 1+4, 1+1+4 and 1+1+1+4, but 1+1+1+2+2...
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and...
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
IN PYHTON CODE Question #3 Produce a function prime_factors.dict that has one integer variable n that...
IN PYHTON CODE
Question #3 Produce a function prime_factors.dict that has one integer variable n that is assumed to be greater than 1. The function will return a dictionary with keys the integers from 2 to n, inclusive, and with values the set of prime factors of each key. So, the command prime_factors.dict (8) will return the dictionary 2 123,3: 3),4 2),5 (53,6 2,3),7:7),8 {2)) Note that even though the prime number 2 appears twice in the prime fac- torization...
A perfect number is a positive integer that is equal to the sum of its (proper)...
A perfect number is a positive integer that is equal to the sum of its (proper) positive divisors, including 1 but excluding itself. A divisor of a number is one which divides the number evenly (i.e., without a remainder). For example, consider number 6. Its divisors are 1, 2, 3, and 6. Since we do not include number itself, we only have 1, 2, and 3. Because the sum of these divisors of 6 is 6, i.e., 1 + 2...
A positive integer greater than 1 is said to be prime if it has no divisors...
A positive integer greater than 1 is said to be prime if it has no divisors other than 1 and itself. A positive integer greater than 1 is composite if it is not prime. Write a program that defines two functions is_prime() and list_primes(). is_prime() will determine if a number is prime. list_primes() will list all the prime numbers below itself other than 1. For example, the first five prime numbers are: 2, 3, 5, 7 and 11." THE PROGRAM...
Please paste your code and a screenshot of your output! 1. An integer n is divisible...
Please paste your code and a screenshot of your output!
1. An integer n is divisible by 9 if the sum of its digits is divisible by 9. Develop a program to determine whether or not the following numbers are divisible by 9: n= 154368 n 621594 n-123456 2. A number is said to be perfect if the sum of its divisors (except for itself) is equal to itself. For example, 6 is a perfect number because the sum of...
A perfect number is a positive integer that equals the sum of all of its divisors...
A perfect number is a positive integer that equals the sum of all of its divisors (including the divisor 1 but excluding the number itself). For example 6, 28 and 496 are perfect numbers because 6=1+2+3 28 1 + 2 + 4 + 7 + 14 496 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Write a program to read a positive integer value, N, and find the smallest perfect number...
Part 15A and 15B (15) Let n E Z+,and let d be a positive divisor of...
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...
Write a program “hw4.c” that reads integer (less than or equal 100) from the keyboard and,...
Write a program “hw4.c” that reads integer (less than or equal 100) from the keyboard and, on the output, writes the sum of the divisors of n (other than itself). For integers less than or equal to 1 it should print 0. For example, the input -3 0 1 4 5 6 12 should generate the output 0 0 0 3 1 6 16 Explanation of output: The input -3 is less than 1, output is 0. The input 0...
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
Challenge activity: A partition of a positive integer n is the expression of n as the sum of positive integers, where order does not matter. For example, two partitions of 7 are 7 1+1+1+4 and 7=1+1+1+2+2. A partition of n is perfect if every integer from 1 to n can be written uniquely as the sum of elements in the partition. 1+1+1+4 is perfect since 1-7 are expressed only as 1, 1+1, 1+1+1, 4, 1+4, 1+1+4 and 1+1+1+4, but 1+1+1+2+2...
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
IN PYHTON CODE
Question #3 Produce a function prime_factors.dict that has one integer variable n that is assumed to be greater than 1. The function will return a dictionary with keys the integers from 2 to n, inclusive, and with values the set of prime factors of each key. So, the command prime_factors.dict (8) will return the dictionary 2 123,3: 3),4 2),5 (53,6 2,3),7:7),8 {2)) Note that even though the prime number 2 appears twice in the prime fac- torization...
Please paste your code and a screenshot of your output!
1. An integer n is divisible by 9 if the sum of its digits is divisible by 9. Develop a program to determine whether or not the following numbers are divisible by 9: n= 154368 n 621594 n-123456 2. A number is said to be perfect if the sum of its divisors (except for itself) is equal to itself. For example, 6 is a perfect number because the sum of...
A perfect number is a positive integer that equals the sum of all of its divisors (including the divisor 1 but excluding the number itself). For example 6, 28 and 496 are perfect numbers because 6=1+2+3 28 1 + 2 + 4 + 7 + 14 496 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Write a program to read a positive integer value, N, and find the smallest perfect number...
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...
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