/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Determine the prime factorizatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 4

Determine the prime factorization for each of the following: a) \(8 !\) b) \(10 !\) c) \(12 !\)

Short Answer

Expert verified
a) The prime factorization of \(8!\) is \(2^7 * 3^2 * 5 * 7\). \nb) The prime factorization of \(10!\) is \(2^8 * 3^4 * 5^2 * 7\). \nc) The prime factorization of \(12!\) is \(2^10 * 3^5 * 5^2 * 7 * 11\).

Step by step solution

01

Calculate the Factorial

Firstly, calculate the factorial of the given numbers. \nUsage of the factorial formula is necessary for this step: \[n! = n*(n-1)*(n-2)*...*3*2*1\]\na) \(8! = 8*7*6*5*4*3*2*1 = 40320\)\nb) \(10! = 10*9*8*7*6*5*4*3*2*1 = 3628800\)\nc) \(12! = 12*11*10*9*8*7*6*5*4*3*2*1 = 479001600\)
02

Determine the Prime Factors

A prime number has no other divisors except 1 and the number itself. To determine the prime factors, start with the smallest prime number, which is 2, and continue with other prime numbers (3, 5, 7, and so on) until the complete factorization is done.\na) Prime factors of 40320: \(2^7 * 3^2 * 5 * 7\)\nb) Prime factors of 3628800: \(2^8 * 3^4 * 5^2 * 7\)\nc) Prime factors of 479001600: \(2^10 * 3^5 * 5^2 * 7 * 11\)
03

Confirm the Factorization

To finish, multiply the prime factors to ensure they equal to the original factorial value.\na) \(2^7 * 3^2 * 5 * 7 = 40320\)\nb) \(2^8 * 3^4 * 5^2 * 7 = 3628800\)\nc) \(2^10 * 3^5 * 5^2 * 7 * 11 = 479001600\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial Calculation
Factorial calculation is a fundamental concept in mathematics. It involves multiplying a sequence of descending natural numbers down to 1. For instance, the factorial of a number "n," denoted as \(n!\), is calculated as \(n \times (n-1) \times (n-2) \cdots \times 3 \times 2 \times 1\). This sequence multiplies every whole number from "n" down to 1.
To illustrate:
  • \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\)
  • \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800\)
  • \(12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479,001,600\)
Factorials are widely used in permutations, combinations, and various mathematical calculations requiring product sequences.
Prime Numbers
Prime numbers are the building blocks of the integers, as every number can be expressed as a product of prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself.
Prime numbers have unique properties:
  • The smallest prime number is 2, which is also the only even prime.
  • Other prime numbers examples include 3, 5, 7, 11, 13, and so on.
  • They are used to determine the prime factorization of any number.
When determining the prime factors of a number, one starts with the smallest prime (2) and divides the number until it cannot be further divided by that prime, then move to the next smallest prime.
Mathematical Proofs
Mathematical proofs are logical arguments that demonstrate the truth of a statement beyond any doubt. They are the foundation upon which the validity of mathematical concepts is established.
Understanding proofs involve:
  • Using deductive reasoning to connect logical steps.
  • Relying on previously established results and axioms.
  • Breaking down complex problems into simpler statements that are easier to validate.
In the context of factorials and prime numbers, a proof might show that a given number is indeed the prime factorization of a factorial by multiplying the prime factors to verify it equates to the original factorial number itself. A deeper understanding of mathematical proofs helps to solidify one's conceptual grasp of the subject matter, ensuring not just that something is true, but why it is true.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine the quotient \(q\) and remainder \(r\) for each of the following, where \(a\) is the dividend and \(b\) is the divisor. a) \(a=23, \quad b=7\) b) \(a=-115, b=12\) c) \(a=0, \quad b=42\) d) \(a=37, \quad b=1\) e) \(a=434, b=31\) f) \(a=-644, b=85\)

For \(n \in \mathbf{Z}^{+}\), prove each of the following by mathematical induction: a) \(5 \mid\left(n^{5}-n\right)\) b) \(6 \mid\left(n^{3}+5 n\right)\)

If \(n \in \mathbf{Z}^{+}\)and \(n \geq 2\), prove that \(2^{n}<\left(\begin{array}{c}2 n \\ n\end{array}\right)<4^{n}\).

Let \(a, b, c, d \in \mathbf{Z}^{+}\). Prove that (a) \([(a \mid b) \wedge(c \mid d)] \Rightarrow a c \mid b d ;\) (b) \(a|b \Rightarrow a c| b c ;\) and, (c) \(a c|b c \Rightarrow a| b\).

a) Determine all \(w, x, y \in \mathbf{Z}\) that satisfy the following system of Diophantine equations. $$ \begin{aligned} &w+x+y=50 \\ &w+13 x+31 y=116 \end{aligned} $$ b) Is there any solution in part (a) where \(w, x, y>0\) ? c) Is there any solution in part (a) where \(w>10, x>28\), and \(y>-15\) ?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.