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Chapter 10: Problem 10

Write each of the following integers as a product of prime numbers. a. 123 b. 375 c. 927

Short Answer

Expert verified
a. \(123 = 3 * 41\) \n b. \(375 = 5^3 * 3\) \n c. \(927 = 3^2 * 103\)

Step by step solution

01

Prime factorization for 123

First, find the smallest prime number that divides 123. We find that 3 is a factor of 123. So we divide 123 by 3 to get 41, which is a prime number itself. Thus, 123 can be represented as \(3 * 41\).
02

Prime factorization for 375

Next, we will find the smallest prime number that divides 375. We see that 5 is a factor of 375. So we divide 375 by 5 to get 75. Again, we divide 75 by 5 to get 15, and lastly, we divide 15 by 5 to get 3 which is a prime number. Thus, 375 can be represented as \(5 * 5 * 5 * 3\) or \(5^3 * 3\).
03

Prime factorization for 927

Lastly, we will find the smallest prime number that divides 927. We find that 3 is a factor of 927. After dividing 927 by 3, we get 309. Now, we divide 309 by 3 again to get 103 which is a prime number itself. Therefore, 927 can be represented as \(3 * 3 * 103\) or \(3^2 * 103\) .

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Key Concepts

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

Integers
Integers are whole numbers; they can be positive, negative, or even zero. In mathematics, integers do not contain fractions or decimals, which makes them very practical for counting and basic arithmetic.

When dealing with integers, it鈥檚 helpful to remember:
  • Positive integers are greater than zero (e.g., 1, 2, 3).
  • Negative integers are less than zero (e.g., -1, -2, -3).
  • Zero is an integer that is neither positive nor negative.
Understanding the properties of integers is essential in operations like addition, subtraction, multiplication, and division. Knowing how to work with integers helps in various mathematical tasks, like finding common denominators or solving equations. Prime factorization, the process of breaking down an integer into the product of prime numbers, is key to understanding the building blocks of numbers.
Product of Primes
The product of primes refers to expressing a number as a multiplication of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They are the building blocks of all numbers.

To write a number as a product of primes, you follow these steps:
  • Find the smallest prime number that divides the integer without a remainder.
  • Divide the integer by this prime number.
  • Continue the process with the result until the final quotient is a prime number.
For example:
  • Consider the number 123. Dividing it by the prime number 3 gives 41, another prime number. Hence, 123 as a product of primes is \(3 \times 41\).
  • The number 375 can be broken down to \(5 \times 5 \times 5 \times 3\) or \(5^3 \times 3\).
This method shows how every integer can be uniquely represented as a product of primes, an essential concept in number theory.
Mathematical Problem Solving
Mathematical problem solving involves breaking down problems into manageable parts and finding solutions using logical reasoning and methodology. Prime factorization is a form of problem-solving where you must systematically decompose a number into its simplest components: prime numbers.

When solving such problems:
  • Identify the problem clearly.
  • Use a step-by-step approach to break the problem into parts.
  • Verify each step for accuracy before proceeding.
Prime factorization exercises, such as breaking down 927 into \(3^2 \times 103\), demonstrate mathematical problem-solving by organizing numbers into more understandable forms. This organized approach helps not only in solving mathematical exercises but also in developing critical thinking and analytical skills for more complex challenges.

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Most popular questions from this chapter

Find the positive divisors of the following integers. a. 72 b. 31 c. 123

Find the necessary condition to have equation\\[m x \equiv m y \bmod n \text { imply } x \equiv y \bmod n\\].

Prove that if \(n \geq m>0,\) then \(\operatorname{gcd}(m, n)=\operatorname{gcd}\) \((m, n-m)\).

Show that if \(G=(S, *)\) is a finite group and \(a \in S\) then there exists integers \(k, m \geq 1\) such that \(a^{k}=a^{k} a^{m}\).

Let \(p\) and \(q\) be two prime numbers. If \(p=q+2\) then \(p\) and \(q\) are called "twin prime numbers." Find two pairs of twin prime numbers.
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