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Chapter 6: Problem 63

Write the prime factored form of each number. $$ 18 $$

Short Answer

Expert verified
The prime factors of 18 are \( 2 \times 3^2 \).

Step by step solution

01

Identify the smallest prime number

The smallest prime number is 2. Check if 2 is a factor of 18.
02

Divide by 2

Since 18 is even, divide 18 by 2 to find the quotient. \( 18 \div\ 2 = 9 \)
03

Identify the next smallest prime number

The next smallest prime number is 3. Check if 3 is a factor of 9.
04

Divide by 3

Since 9 can be divided by 3, divide 9 by 3 to find the quotient. \( 9 \div\ 3 = 3 \)
05

Divide by 3 again

Perform the division again since the quotient 3 is also divisible by 3. \( 3 \div\ 3 = 1 \)
06

Write the prime factors

List the prime numbers that were used in the division steps. Prime factors of 18: \( 2 \times 3 \times 3 \) or \( 2 \times 3^2 \).

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Key Concepts

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

Prime Numbers
Prime numbers are the basic building blocks of all numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
In other words, a prime number has exactly two distinct positive divisors: 1 and itself. Examples of prime numbers are:
  • 2
  • 3
  • 5
  • 7
  • 11
Notice 2 is the smallest and the only even prime number. Understanding primes is crucial because every whole number can be expressed as a product of primes, a process called prime factorization.
Divisibility
Divisibility is the concept of one number being divisible by another without leaving a remainder.
For example, 18 is divisible by 2 because when you divide 18 by 2, you get exactly 9 with no remainder.
Divisibility rules can help you quickly determine if one number is divisible by another. Here are a few rules for smaller prime numbers:
  • 2: A number is divisible by 2 if it is even.
  • 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • 5: A number is divisible by 5 if it ends in 0 or 5.
Knowing these rules will help you identify the prime factors of a number more easily.
Factorization Step-by-Step
Prime factorization is the process of breaking down a composite number into the prime numbers that multiply to form it. Here is the step-by-step process using the example of 18:
  • Step 1: Identify the smallest prime number, which is 2. Check if 2 is a factor of 18.
  • Step 2: Since 18 is even, divide it by 2: \(18 \div 2 = 9\).
  • Step 3: Identify the next smallest prime number, which is 3. Check if 3 is a factor of 9.
  • Step 4: Since 9 can be divided by 3, divide 9 by 3: \(9 \div 3 = 3\).
  • Step 5: Perform the division again since the quotient 3 is also divisible by 3: \(3 \div 3 = 1\).
  • Step 6: List the prime numbers used in these steps: Prime factors of 18: \[18 = 2 \times 3 \times 3\ or 2 \times 3^2\]
This method ensures you identify all the prime factors in an organized way.

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Most popular questions from this chapter

Write the prime factored form of each number. $$ 60 $$

Multiply. Write each answer in lowest terms. $$ \frac{(x-y)^{2}}{2} \cdot \frac{24}{3(x-y)} $$

Match each division problem in Column I with the correct quotient in Column II. (a) \(\frac{5 x^{3}}{10 x^{4}} \div \frac{10 x^{7}}{4 x}\) (b) \(\frac{10 x^{4}}{5 x^{3}} \div \frac{10 x^{7}}{4 x}\) (c) \(\frac{5 x^{3}}{10 x^{4}} \div \frac{4 x}{10 x^{7}}\) (d) \(\frac{10 x^{4}}{5 x^{3}} \div \frac{4 x}{10 x^{7}}\) A. \(\frac{5 x^{5}}{4}\) B. \(5 x^{7}\) C. \(\frac{4}{5 x^{5}}\) D. \( \frac{1}{5 x^{7}}\)

Multiply or divide. Write each answer in lowest terms. $$ \frac{(q-3)^{4}(q+2)}{q^{2}+3 q+2} \div \frac{q^{2}-6 q+9}{q^{2}+4 q+4} $$

Multiply. Write each answer in lowest terms. $$ \frac{12 m^{5}}{-2 m^{2}} \cdot \frac{6 m^{6}}{28 m^{3}} $$
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