The prime factorization of a positive integer n is p^3. Which of the following is true?...
The prime factorization of a positive integer n is p^3. Which of the following is true? Explain and show your answers.
I. n cannot be even
II. n has only one positive prime factor.
III, n has exactly three distinct factors.
Homework Answers
Solution:
The prime factorization of a positive integer 
is 
, where 
is a prime number. So all possible factor of
are 
(i) If we consider 
, then the prime factorization of 
is 
for a prime 
, but
is even.
(ii) Clearly
has only prime factor 
The other factors are not prime.
(iii) If
has three distinct factors then the prime factorization of
cannot be
for any prime
. So
cannot has exactly three distinct factors.
Therefore option (ii) is correct.
Add Answer to:
The prime factorization of a positive integer n is p^3. Which of
the following is true?...
Find the smallest positive integer that has precisely n distinct prime divisors. 'Distinct prime divisor'Example: the...
Find the smallest positive integer that has precisely n distinct prime divisors. 'Distinct prime divisor'Example: the prime factorization of 8 is 2 * 2 * 2, so it has one distinct prime divisor. Another: the prime factorization of 12 is 2 * 2 * 3, so it has two distinct prime divisors. A third: 30 = 2 * 3 * 5, which gives it three distinct prime divisors. (n = 24 ⇒ 23768741896345550770650537601358310. From this you conclude that you cannot...
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive...
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive integers n such that n|10! and gcd(n, 27.34.7) = 27.3.7. Justify your answer. Question 4 Let m, n E N. Prove that ged(m2, n2) = (gcd(m, n))2. Question 5 Let p and q be consecutive odd primes with p < q. Prove that (p + q) has at least three prime divisors (not necessarily distinct).
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n)....
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
Consider the following statements concerning a positive integer : (i.) if n is a multiple of...
Consider the following statements concerning a positive integer : (i.) if n is a multiple of 3, then n2 is a multiple of 3 [ii.) if n' is a multiple of 12, then n is a multiple of 12 (iii.) if nis a multiple of 6, then nis a multiple of 6 Which of the statements are true? Select one: O i. and iii. O All the statements None of the statements Only ii.
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...
Use python for programming the fundamental theorem of arithmetic (single factorization theorem), which affirms that every...
Use python for programming the fundamental theorem of arithmetic (single factorization theorem), which affirms that every positive integer greater than 1 is a prime number or a single product of prime numbers. Show the factors in a list and show a dictionary where the keys are the factors of the number entered and the values are how many times each factor appears in the unique combination.
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...
6. Let n be any positive integer which n = pq for distinct odd primes p....
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
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=...
Let n be a positive integer. Show that nº + 4n +5 has no prime divisor...
Let n be a positive integer. Show that nº + 4n +5 has no prime divisor p with p 3 mod 4.
Question 3 (a) Write down the prime factorization of 10!. (b) Find the number of positive integers n such that n|10! and gcd(n, 27.34.7) = 27.3.7. Justify your answer. Question 4 Let m, n E N. Prove that ged(m2, n2) = (gcd(m, n))2. Question 5 Let p and q be consecutive odd primes with p < q. Prove that (p + q) has at least three prime divisors (not necessarily distinct).
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
Consider the following statements concerning a positive integer : (i.) if n is a multiple of 3, then n2 is a multiple of 3 [ii.) if n' is a multiple of 12, then n is a multiple of 12 (iii.) if nis a multiple of 6, then nis a multiple of 6 Which of the statements are true? Select one: O i. and iii. O All the statements None of the statements Only ii.
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...
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...
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
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=...
Let n be a positive integer. Show that nº + 4n +5 has no prime divisor p with p 3 mod 4.
Most questions answered within 3 hours.
-
From the literature (reference your sources): What are the
lattice parameters of calcite and aragonite? Why...
asked 29 minutes ago -
Your system is rejecting the question am asking which is
preceded by a case study. It...
asked 33 minutes ago -
3. On January 2, 2000, Larry creates a trust with himself as
trustee. Larry as trustee...
asked 30 minutes ago -
A member of the volleyball team spikes the ball. During this
process, she changes the velocity...
asked 37 minutes ago -
Are adult gamers less likely to use a gaming console (Xbox,
PlayStation, Wii, etc...) than teen...
asked 1 hour ago -
The University of
Texas recently reported that 43% of college students aged 18-24
would spend their...
asked 1 hour ago -
The length of stay at a specific emergency department in
Phoenix, Arizona, in 2009 had a...
asked 57 minutes ago -
. Please give the mechanism for this type of problem. Step by
Step
The toxin that...
asked 1 hour ago -
If you have a 1M stock solution and you want to dilute 1 :10
with water,...
asked 1 hour ago -
In a load instruction, the effective address is obtained by
A) Retriving the address from a...
asked 1 hour ago -
Use the following information to answer this question.
Windswept, Inc. 2017 Income Statement ($ in millions)...
asked 1 hour ago -
A mutual fund salesperson has arranged to call on four people
tomorrow. Based on past experience...
asked 1 hour ago