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Chapter 5: Problem 29

Find the prime factorization of each composite number. 105

Short Answer

Expert verified
The prime factorization of 105 is \(105 = 5 \times 3 \times 7\).

Step by step solution

01

Divide by smallest prime number

Start dividing the composite number 105 by the smallest prime number, which is 2. As 105 is not divisible by 2, move to the next prime number.
02

Divide by next smallest prime number

Try dividing 105 by the next smallest prime number which is 3. As 105 is not divisible by 3, move to the next prime number.
03

Divide by next smallest prime number

Divide 105 by the next smallest prime number which is 5. 105 is divisible by 5 and the quotient is 21.
04

Continue the process with the quotient

Continue the same process with the quotient 21. Start by dividing it with the smallest prime number 2. As it is not divisible, move to the next prime number 3. 21 is divisible by 3 and the quotient is 7. Now, the quotient 7 is a prime number.
05

Write down the prime factorization

The prime numbers used to divide 105 are 5, 3 and 7. Hence, the prime factorization of 105 can be written as \(105 = 5 \times 3 \times 7\).

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Key Concepts

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

Composite Numbers
Understanding composite numbers is key in prime factorization. A composite number is a natural number greater than one that is not prime. This means it can be divided exactly by numbers other than one and itself. Let's explore this idea further:
  • Composite numbers have more than two factors.
  • For example, 105 is a composite number because it has factors like 1, 3, 5, 7, 15, 21, 35, and 105.
  • Prime factorization is used to express a composite number as a product of prime numbers.
Recognizing composite numbers is crucial because it sets the stage for breaking them down into their prime building blocks.
Divisibility
Divisibility helps you determine which numbers can divide another number without leaving a remainder. This concept is essential when finding prime factors:
  • To check for divisibility, you often start with the smallest prime numbers.
  • A number is divisible by another if the division results in a whole number.
  • For instance, 105 is not divisible by 2 (as it's odd), but it is divisible by 3 and 5.
Understanding divisibility rules simplifies the task of identifying which prime numbers can divide a given composite number.
Prime Numbers
Prime numbers are the foundation of prime factorization. A prime number is a natural number greater than 1, with no divisors other than 1 and itself. Consider these facts:
  • Prime numbers like 2, 3, 5, and 7 are integral in building other numbers.
  • The only factors of a prime number are one and the number itself.
  • Prime factorization involves expressing a composite number like 105 as a product of prime numbers: 5, 3, and 7.
Understanding primes and their properties makes it easier to factorize composite numbers precisely.
Mathematical Procedures
Mathematical procedures provide a structured approach for tackling problems like prime factorization. Here's how you can follow these steps efficiently:
  • Begin with the smallest prime number and test for divisibility.
  • If divisible, write down the prime factor and divide the number by the prime.
  • Repeat the process with the quotient, using the next smallest prime number.
  • Continue until the quotient is a prime number itself.
This systematic process ensures you break down composite numbers into their prime components accurately. For example, starting with 105, you divide by 5, then by 3, and finally recognize that 7 is prime. Therefore, 105 is expressed as a product of these primes: \(105 = 5 \times 3 \times 7\). It's all about meticulously following the steps to reach the correct factorization.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-\frac{1}{16}, r=-4\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{8}\), when \(a_{1}=1,000,000, r=0.1\).

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(3,-6,12,-24, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{5}, r=\frac{1}{2}\)

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to \(2015 .\) Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,18.4 \%\) of American women ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.6\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American women ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American women ages 25 and older who will be college graduates by 2029 .
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