Problem 76 - Elementary Algebra

Question

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 60 a b ; 36 x y $$

Step-by-Step Solution

Answer

The LCM of 60ab and 36xy is 180abxy.

Step 1: Prime Factorization of 60ab

First, find the prime factors of 60. \[ 60 = 2^2 \times 3 \times 5 \] Since the expression is 60ab, the full factorization is: \[ 60ab = 2^2 \times 3 \times 5 \times a \times b \]

Step 2: Prime Factorization of 36xy

Next, find the prime factors of 36. \[ 36 = 2^2 \times 3^2 \] Since the expression is 36xy, the full factorization is: \[ 36xy = 2^2 \times 3^2 \times x \times y \]

Step 3: Identify Common and Highest Power Factors

Combine the prime factors of both expressions and choose the highest powers of each prime factor. For 60ab: \[ 2^2 \times 3 \times 5 \times a \times b \] For 36xy: \[ 2^2 \times 3^2 \times x \times y \] List all primes with their highest powers, including all the unique variables: \[ 2^2, 3^2, 5, a, b, x, y \]

Step 4: Calculate the Least Common Multiple (LCM)

Multiply the highest powers of all prime factors and variables: \[ \text{LCM} = 2^2 \times 3^2 \times 5 \times a \times b \times x \times y \] Hence, the LCM of 60ab and 36xy is: \[ \text{LCM} = 180abxy \]

Key Concepts

Prime Factorization

Prime factorization is an essential step for solving least common multiple (LCM) problems. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Some examples are 2, 3, 5, and 7. To find the prime factorization of a number, we repeatedly divide the number by its smallest prime factor. For example, for the number 60, the process is as follows: 60 can be divided by 2: \[ 60 \div 2 = 30 \] 30 can again be divided by 2: \[ 30 \div 2 = 15 \] 15 can be divided by 3: \[ 15 \div 3 = 5 \] And 5 is already a prime number. So, the prime factorization of 60 is: \[ 60 = 2^2 \times 3 \times 5 \] This process is crucial because it allows us to break down numbers into their building blocks, making it easier to compute the LCM.

LCM Calculation

The Least Common Multiple (LCM) of two numbers is the smallest multiple that both numbers share. After prime factorization, we combine the factors, taking the highest power of each factor. For the numbers represented in the exercise, 60ab and 36xy: - Factorize 60ab to get: \[ 60ab = 2^2 \times 3 \times 5 \times a \times b \] - Factorize 36xy to get: \[ 36xy = 2^2 \times 3^2 \times x \times y \] To calculate the LCM, we take the highest power of each prime and factor present: - Highest power of 2: \[ 2^2 \] - Highest power of 3: \[ 3^2 \] - Factor 5: \[ 5 \] - Variables a, b, x, and y as they appear in the expressions. Thus, the LCM is: \[ \text{LCM} = 2^2 \times 3^2 \times 5 \times a \times b \times x \times y \] Combining all these, the LCM equals 180abxy.

Highest Power Factors

When calculating the LCM, choosing the highest power of each factor ensures that the LCM is indeed the smallest multiple common to both numbers. Let's revisit our previous prime factorizations: For 60ab: \[ 2^2 \times 3 \times 5 \times a \times b \] For 36xy: \[ 2^2 \times 3^2 \times x \times y \] In the LCM calculation, for each unique factor, we need to pick the highest number of times it appears in any number. - Factor 2 appears as \[ 2^2 \] in both. - Factor 3 appears as \[ 3^2 \] in 36xy and as \[ 3 \] in 60ab, so we choose \[ 3^2 \].
- Factor 5 appears only once in 60ab.
- Variables a, b, x, and y each appear as themselves. This careful selection ensures all factors are fully represented, without repeating any more than needed, making the LCM the efficient smallest common multiple.

Variables in Algebra

Variables play a crucial role in algebra, representing unknown or generalized numbers. In the given exercise, 'a', 'b', 'x', and 'y' are variables. They act like placeholders and can stand for any number. When calculating the LCM in algebraic terms, these variables are treated just like any other prime factor. For the expressions 60ab and 36xy, the variables 'a', 'b', 'x', and 'y' must be included in the LCM because:

• 'a' and 'b' appear in the term 60ab
• 'x' and 'y' appear in the term 36xy

Therefore, in the final LCM calculation, these variables are added in the expression without raising them to any specific power as they already represent unique entities in the problem. This results in the LCM being expressed as 180abxy, systematically covering the presence of all prime factors and associated variables.