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Chapter 5: Problem 25

In Exercises 25-44, find the prime factorization of each composite number. 75

Short Answer

Expert verified
The prime factorization of 75 is \(75 = 5^2 \times 3^1\).

Step by step solution

01

Identify the Smallest Prime Number

The smallest prime number is 2, but 75 is an odd number so we start with the next prime number which is 3, but it cannot divide 75 evenly. The next prime number is 5.
02

Divide the Composite Number

Now we have to divide 75 by 5. This gives us a quotient of 15. The number 15 is not a prime number, so we continue the process.
03

Further Divide the Quantity

Next, we again divide 15 by 5. This gives us a quotient of 3 which is a prime number. We stop as we cannot further divide a prime number.
04

Writing the Prime Factorization

The prime factors of 75 are 5 two times and 3 one time. We express this as \(75 = 5^2 \times 3^1\).

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Key Concepts

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

Composite Numbers
Composite numbers are whole numbers that have more than two factors. Unlike prime numbers, which only have 1 and itself as factors, composite numbers can be divided evenly by numbers other than 1 and itself. This makes them interesting in mathematics, especially when learning about different kinds of numbers.
  • For example, the number 75 is a composite number because it is divisible by 1, 3, 5, 15, 25, and 75.
  • Composite numbers are essentially made up of smaller numbers, and breaking them down can help us find their prime factors.
Understanding composite numbers is key when trying to find prime factorizations. This is the process of breaking down a composite number into a product of prime numbers.
Prime Numbers
Prime numbers are the building blocks of all numbers. A prime number is a number greater than 1 that cannot be divided evenly by any other numbers except for 1 and itself. This unique characteristic makes them essential in the factorization process.
  • The smallest prime number is 2, which is also the only even prime number.
  • Other examples of prime numbers include 3, 5, 7, and 11.
  • Prime numbers cannot be broken down any further, hence once we reach them in the factorization process we stop.
Identifying these numbers is essential when performing prime factorization. We use prime numbers like tools to split a composite number into smaller parts until only prime numbers remain.
Division
Division is a fundamental operation in mathematics that involves determining how many times one number is contained within another. When it comes to prime factorization, division plays a critical role in breaking down composite numbers.
The process of division in prime factorization is repetitive. First, you find a prime number that can divide your composite number evenly, then perform the division:
  • Start with the smallest possible prime number to see if it divides the composite number without leaving a remainder.
  • If it does not divide evenly, try the next smallest prime number.
  • Continue this process until the quotient is a prime number, indicating that you can no longer divide further.
This step-by-step division highlights why division is a logical step. It's through continuous division that we find the prime factors, allowing us to express the composite number as a product of these primes.

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Most popular questions from this chapter

You are offered a job that pays \(\$ 50,000\) for the first year with an annual increase of \(3 \%\) per year beginning in the second year. That is, beginning in year 2 , your salary will be \(1.03\) times what it was in the previous year. What can you expect to earn in your sixth year on the job? Round to the nearest dollar.

Convert 365 days (one year) to hours, to minutes, and, finally, to seconds, to determine how many seconds there are in a year. Express the answer in scientific notation.

Explain how to add integers.

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. \(4,-12,36,-108, \ldots\)

Use properties of exponents to simplify each expression. First express the answer in exponential form. Then evaluate the expression. \(\frac{4^{7}}{4^{5}}\)
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