Problem 34 - Elementary Algebra

Question

Multiply and reduce to lowest terms. $$ 34 \cdot 209 $$

Step-by-Step Solution

Answer

The product is 7106, already in simplest form for multiplication.

Step 1: Perform the Multiplication

First, multiply the numbers 34 and 209. This can be done by setting it up as follows:\[34 \times 209 = 7106\]

Step 2: Check for Simplification

Now that we have the product 7106, we need to check if this number can be reduced to lowest terms. Simplification applies to fractions, so if the exercise intended to reduce a product to relatively prime factors, we would need to find the greatest common divisor (GCD) of the factors involved.

Step 3: Identify Prime Factors and Simplify if Needed

Since 34 and 209 are integers and their product is not in fractional form, simplification doesn't directly apply here as with fractions. Instead, we should verify if 7106 can be expressed through simpler factors. Decompose 34 into 2 and 17, and 209 as 11 and 19: - 34 = 2 × 17 - 209 = 11 × 19 These prime factorizations do not share common factors, so they cannot simplify further under multiplication terms.

Key Concepts

Prime Factorization

Prime factorization is a method of expressing a number as a product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To find the prime factorization of a number, you start dividing by the smallest prime, which is 2, and move on to larger primes like 3, 5, 7, etc., until you can't divide anymore without leaving a remainder.

For example:

• 34 can be split as 2 × 17.
• 209 can be split as 11 × 19.

Notice that these prime numbers multiplied together give the original number. Prime factorization is particularly useful in finding the greatest common divisor of numbers or simplifying fractions. It's like breaking down a complex problem into its simplest elements. Always ensure the numbers you are using are indeed prime and multiply to get back the original number.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both integers without leaving a remainder. To find the GCD using prime factorization, you identify the common prime factors in both numbers and then multiply them together.

In our example, the numbers 34 and 209 do not share common prime factors besides the "1", so their GCD is 1. This essentially means that 34 and 209 are coprime, or relatively prime, meaning they do not reduce further to simpler terms through division by any number other than 1.

Steps to find GCD:

• Identify the prime factors of each number.
• Find the common prime factors.
• Multiply the common factors if any exist. If none, the GCD is 1.

The GCD is the key concept when reducing fractions, as it helps simplify to the lowest possible terms.

Simplification of Integers

Simplifying integers is about breaking them down to their simplest form, often by identifying any shared factors. However, in multiplication-only problems, like finding the product of 34 and 209, simplification specifically pertains to identifying prime factors and reducing based on common elements. For products that are not fractions, this typically translates to understanding the components of the product itself, rather than "simplifying" in a traditional fraction sense.

When we multiplied 34 by 209:

• The product 7106 was the result.
• We then examined its prime factors: 34 as 2 × 17 and 209 as 11 × 19.

Since there were no overlapping factors, 7106 is already in its simplest factored form. This process of acknowledgment assures us that no further simplification is needed unless dealing with ratios or fractions. For further clarity, simplification of integers in a math sense generally refers to breaking them down to their base components for ease of use in calculations or problem-solving.