Problem 14 - Algebra for College Students

Question

Factor each of the composite numbers into the product of prime numbers. For example, \(30=2 \cdot 3 \cdot 5 .\) $$49$$

Step-by-Step Solution

Answer

49 is factored into primes as \(7 \times 7\).

Step 1: Identify the Number

We start with the number 49 and aim to express it as a product of prime numbers.

Step 2: Find a Prime Factor

Identify one prime number that divides the given number evenly. Since 49 is a perfect square, check if it is divisible by small prime numbers like 2, 3, 5, and so on. We find that 49 is not divisible by these but is divisible by 7.

Step 3: Divide and Simplify

Divide 49 by the prime number 7: \[49 \div 7 = 7\]This tells us that 49 can be expressed as 7 times another number, and the quotient is 7, which is also a prime number.

Step 4: Express as a Product of Primes

We have found the prime factors of 49. It can be expressed as a product of the same prime number: \[49 = 7 \times 7\]Therefore, the factorization of 49 into prime factors is complete.

Key Concepts

Composite Numbers

Composite numbers are fascinating and quite different from prime numbers. Unlike primes, composite numbers have more than two factors, which means that they can be divided evenly by numbers other than 1 and themselves. To understand it better, consider a number like 8. You can divide 8 evenly by not only 1 and 8 but also by 2 and 4. This property makes 8 a composite number. Always remember when working with composite numbers:

• They have at least one divisor other than 1 and themselves.
• They can be "factored" or broken down into smaller whole numbers, specifically prime numbers.
• Every composite number is a product of prime numbers. This is called the prime factorization."

Understanding composite numbers is crucial for performing tasks like factorization, as seen in the context of the original exercise with the number 49.

Perfect Square

A perfect square is a special type of number that is the result of multiplying a number by itself. It’s like having a number "squared". In more technical terms, if you have a number n, then n squared ( ^2) is a perfect square. For instance, 49 is a great example. It's the product of 7 times 7. So, 49 is not only a composite number but also a perfect square. Some interesting features of perfect squares include:

• They always have an odd number of total factors.
• The square root of a perfect square is always an integer.
• Perfect squares grow rapidly as numbers increase, like 1, 4, 9, 16, 25, and so on.

Recognizing perfect squares, like 49, can simplify factorization because you know immediately that its factors are identical.

Prime Numbers

Prime numbers are the "building blocks" of all numbers. A prime number has exactly two distinct positive divisors: 1 and itself. In simpler terms, it can only be divided by 1 and its own value without leaving a remainder. Consider the number 7. It’s a prime number because you can't divide it evenly by any number other than 1 and 7. Here's what to keep in mind about prime numbers:

• They are greater than 1. The smallest prime number is 2, which is also the only even prime.
• Prime numbers play a key role in prime factorization, as every whole number greater than 1 is either a prime or can be expressed as a product of prime numbers.
• They are essential for many branches of mathematics, including cryptography and number theory.

When working on factorizing composites, prime numbers are the tools you use, as illustrated with the number 49 being expressed as 7 times 7.