/*! 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 97 Is \(2 \cdot 3 \cdot 5 \cdot 7 \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 97

Is \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11+7\) a prime or a composite number? Defend your answer.

Short Answer

Expert verified
2317 is a composite number because it is divisible by 7.

Step by step solution

01

Prime Number Definition

First, recall that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
02

Calculate the Expression

Calculate the expression given: \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 + 7\). First, find the product of the prime numbers: \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310\). Adding 7 gives \(2310 + 7 = 2317\).
03

Check Divisibility by 7

Since the original expression added 7 to a multiple of 7, check if 2317 is divisible by 7. Divide 2310 by 7 and verify that it gives a whole number. Therefore, 2317 is divisible by 7.
04

Conclude with Prime or Composite

Since 2317 has a divisor other than 1 and itself (it is divisible by 7), 2317 is a composite number.

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.

Composite Numbers
A number is called composite if it has more than two distinct positive divisors. This means that, unlike prime numbers, composite numbers can be divided exactly by numbers other than 1 and themselves. When working with composite numbers, you can think of them as numbers that can be "broken down" into smaller numbers. For example, 2317 is a composite number.
  • It can be divided not only by 1 and 2317, but also by 7.
  • Having divisors other than 1 and itself confirms its composite nature.
Understanding whether a number is composite helps in many mathematical areas, including simplifying fractions and finding common denominators.
Divisibility
Divisibility is a concept that helps to determine if one number can be divided by another without leaving any remainder. For example, a number is divisible by 7 if, after division, the result is a whole number.
  • To test if a number like 2317 is divisible by 7, you perform division.
  • If 2317 divided by 7 equals an integer, then it is divisible by 7.
In this exercise, 2317 is shown to be divisible by 7, making divisibility a useful tool for understanding why a number is composite. It's a fundamental part of number theory and helps with tasks like simplifying expressions and verifying number properties.
Prime Factorization
Prime factorization involves expressing a composite number as a product of prime numbers. This process breaks down composite numbers into their basic building blocks.
  • Start by dividing the number by the smallest prime number.
  • Continue dividing the result by prime numbers until you reach a prime number.
For example, while the initial part of the exercise dealt with prime numbers up to 11, the resulting composite number, 2317, can be further factorized through divisibility by 7. Knowing how to perform prime factorization is quite practical鈥攊t simplifies solving mathematical problems and understanding number structures.

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

Use this approach to factor Problems \(104-109\). $$(x-3)^{2}+10(x-3)+24$$

How can you determine that \(x^{2}+5 x+12\) is not factorable using integers?

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$25 n^{2}+64$$

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$7 x^{2}+62 x-9=0$$

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$n(n+1)=182$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.