Problem 74 - Elementary Algebra

Question

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 21 x^{3} y^{5} ; 84 x y^{2} $$

Step-by-Step Solution

Answer

84 x^3 y^5

Step 1: Find Prime Factorization of 21

The prime factorization of 21 is: \[ 21 = 3 \times 7 \]

Step 2: Find Prime Factorization of 84

The prime factorization of 84 is: \[ 84 = 2^2 \times 3 \times 7 \]

Step 3: Determine the Highest Powers of Each Prime Factor

Identify the highest power of each prime factor in the factorizations of 21 and 84. Prime factors are 2, 3, and 7. The highest powers are: \[ 2^2, 3^1, 7^1 \]

Step 4: Find the Least Common Multiple of Numerical Part

The least common multiple (LCM) of the numerical part 21 and 84 is: \[ LCM = 2^2 \times 3 \times 7 = 84 \]

Step 5: Find LCM of Variable Part

For the variable part, use the highest power of each variable found in the terms: \[ x^3, y^5 \]

Step 6: Combine Results

Combine the numerical LCM and the variable parts to get the final LCM: \[ LCM = 84 x^3 y^5 \]

Key Concepts

prime factorization

Prime factorization is the process of breaking down a composite number into a product of its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
To find the prime factorization, you can keep dividing the number by the smallest possible prime number until you are left with prime numbers.

For example, in this exercise, the prime factorizations were found as follows:
1. For 21: We divide by 3 (smallest prime factor of 21), then by 7. This gives: \(21 = 3 \times 7\)
2. For 84: We first divide by 2 (smallest prime factor of 84), then again by 2. Next, we divide by 3, and finally by 7. This gives: \(84 = 2^2 \times 3 \times 7\)

highest powers

When finding the least common multiple (LCM), it is crucial to find the highest powers of all prime factors involved. This ensures that the LCM is divisible by each original number.
Here’s how to determine the highest powers:
1. List out all prime factors from both numbers.
2. For each prime factor, use the highest exponent found in the factorizations.
In this example, the prime factors are 2, 3, and 7. Their highest powers are:\[2^2, 3^1, \text{and} 7^1\]. These are the values we use in calculating the LCM for the numerical part.

variables in LCM

The Least Common Multiple (LCM) also includes the variable parts in algebraic expressions. To include variables, follow the same steps for prime factors:
1. Identify all variables in the expressions.
2. For each variable, find the highest power present.
In our example, the expressions have variables \(x\) and \(y\):

• From \( x^3 y^5 \), we have \( x^3 \) and \( y^5 \).
• From \( 84xy^2 \), we have \( x \) and \( y^2 \).

We take the highest power of each variable:\[ x^3 \text{ (from } x^3)\text{ and } y^5 \text{ (from } y^5)\].
Combining these results with the numerical LCM gives us the final LCM: \( 84 x^3 y^5 \).