Problem 102 - Intermediate Algebra

Question

2 because 9 is a perfect square. $$ 45 $$ # Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as 9 # 2 because 9 is a perfect square. $$ 45 $$

Step-by-Step Solution

Answer

45 can be written as \( 9 \times 5 \), with 9 being a perfect square.

Step 1: Determine the Factors of the Number

First, factorize the number 45 into its basic components. Start by factoring out the smallest prime number, which is 3: \( 45 \div 3 = 15 \). Thus, 45 can be expressed as \( 3 \times 15 \). Continue factoring 15: \( 15 \div 3 = 5 \). Hence, \( 45 = 3 \times 3 \times 5 \), which is \( 45 = 9 \times 5 \).

Step 2: Identify Perfect Square

Recognize that from the factorization \( 9 \times 5 \), 9 is a perfect square because \( 3^2 = 9 \). Thus, you can express 45 as a product of a perfect square and another integer.

Step 3: Express in Required Form

The task is to write the number 45 as a product of a perfect square and another integer. From the previous step, the expression is \( 45 = 9 \times 5 \), with 9 being the perfect square. Thus, the answer to the exercise is \( 9 \times 5 \).

Key Concepts

Perfect Squares

A perfect square is a number that has an integer as its square root. For example, if you take the number 9 and find its square root, which is 3, you’ll notice that 3 is an integer (a whole number without fractions). This means 9 is a perfect square because it's the result of multiplying the integer 3 by itself.

Recognizing perfect squares is key in problems involving integer factorization. Numbers like 1, 4, 9, 16, 25, and so on are all perfect squares. Each of these numbers can be expressed as the square of an integer:

• 1 = 1²
• 4 = 2²
• 9 = 3²
• 16 = 4²
• 25 = 5²

When dealing with numbers like 45 in exercises, you can look for a factorization that includes a perfect square. This allows you to express the number as a product of a perfect square and another integer, which is what exercises like these often require.

Integer Factorization

Integer factorization involves breaking down a number into its simplest building blocks, which are prime numbers. This process is incredibly useful because it unveils all the basic components of a number. For example, let's take the number 45.

To factorize 45, start by checking the smallest prime number: 2. Since 45 is odd, it is not divisible by 2. Proceed to the next smallest prime number, 3:

• 45 divided by 3 is 15
• 15 divided by 3 is 5

Thus, 45 can be expressed as 3 multiplied by 3 multiplied by 5: \[45 = 3 \times 3 \times 5\]Re-grouping the factors gives you: \[45 = 9 \times 5\]By recognizing 9 as a perfect square (since \(3^2 = 9\)), you successfully factor the number as a product of a perfect square and another integer. Factorization is crucial in simplifying complex problems and finding solutions like this example.

Prime Numbers

Prime numbers are the fundamental building blocks of all natural numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. This means a prime number can only be divided evenly by 1 and itself without leaving a remainder.

Understanding what prime numbers are will aid in the process of integer factorization. Common prime numbers include 2, 3, 5, 7, 11, and so on. When dealing with factorization, starting with the smallest prime numbers helps to break down larger numbers into their prime components.

Let's see how we use primes in factorization with our example of number 45:

• Start dividing by the smallest prime, which is 3.
• 45 divided by 3 gives 15.
• Continue dividing by 3 (since still divisible), and 15 divided by 3 gives 5.
• The result, 5, is a prime number.

Hence, 45 is broken down into prime numbers: \[3 \times 3 \times 5\].

By using prime numbers for factorization, you can reveal the composition of a number. This helps in solving exercises that require expressing numbers in specific forms, like those that include perfect squares.