Problem 32 - Elementary Algebra

Question

Multiply and reduce to lowest terms. $$ 15 \cdot 48 $$

Step-by-Step Solution

Answer

The product is 720, already in its lowest terms.

Step 1: Multiply the Numbers

First, multiply the two numbers. We have 15 multiplied by 48.Calculate this:\[15 imes 48 = 720\]The product of 15 and 48 is 720.

Step 2: Reduce to Lowest Terms

Next, reduce the product to its lowest terms by dividing by the greatest common divisor (GCD).Find the factors of 720. The prime factorization is:\[720 = 2^4 imes 3^2 imes 5\]Since this is not a fraction, 720 itself is at its lowest term as a whole number.

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its simplest building blocks, known as prime numbers. Prime numbers are numbers greater than 1 that can only be divided evenly by 1 and themselves. Some common prime numbers include 2, 3, 5, 7, and 11.
For a number like 720, the goal of prime factorization is to express it as a product of these smallest possible prime numbers.
First, start with the smallest prime number, 2, and divide 720 by 2. Repeat this step until the result is no longer divisible by 2:
- 720 divided by 2 is 360,- 360 divided by 2 is 180,- 180 divided by 2 is 90,- 90 divided by 2 is 45.
Since 45 isn't divisible by 2, the next prime number is 3. Divide 45 by 3:- 45 divided by 3 is 15,- 15 divided by 3 is 5.
The number 5 is a prime number, so we stop here.
Thus, the prime factorization of 720 is:\[720 = 2^4 \times 3^2 \times 5\].
This step is crucial for problems involving divisors because it helps identify all of the number's possible factors using its basic prime components.

Greatest Common Divisor

The greatest common divisor (GCD) of two or more numbers is the largest number that evenly divides all of them. It's useful in reducing fractions to their lowest terms. To find the GCD, one useful approach is to look at the prime factorizations of the numbers.
For example, let's say we want to find the GCD of 36 and 48. Start by writing their prime factorizations:

• 36 = 2^2 \times 3^2
• 48 = 2^4 \times 3^1

Identify the common prime numbers and select the smallest power of each that appears in both factorizations:

• The common prime is 2, with the smallest power being 2^2.
• Another common prime is 3, with the smallest power being 3^1.

Multiply these together to get the GCD:\[GCD = 2^2 \times 3^1 = 12\]
Finding the GCD via prime factorization is a handy method because it gives a thorough understanding of how numbers relate to each other through their factors.

Reducing to Lowest Terms

Reducing a fraction or a number to its lowest terms means simplifying it so that it cannot be divided further while remaining a rational number. While this concept often relates to fractions, it can also apply to whole numbers if they are presented as products (or multiples) of other whole numbers.
For example, let's consider the product \(15 \times 48 = 720\). Even though 720 is not a fraction, this does not automatically exclude it from being reduced. However, for fractions, use their GCD to divide both the numerator and the denominator:

Consider a fraction like 48/360:

• First, find the GCD, which is 12.
• Divide both 48 and 360 by 12.
• You'll get 4/30.
• Notice 4/30 can be further divided by 2, reducing it to 2/15.

Thus, the fraction 48/360 in its lowest form is 2/15.
Reducing numbers to their lowest terms simplifies the problem you are dealing with and makes any further arithmetic much more straightforward.