Question

The following are the steps involved in finding the LCM of 72 and 48 by prime factorization method. Arrange them in sequential order from first to last. (a) \(72=2^{3} \times 3^{2}\) and \(48=2^{4} \times 3^{1}\) (b) \(\mathrm{LCM}=2^{4} \times 3^{2}\) (c) All the distinct factors with highest exponents are \(2^{4}\) and \(3^{2}\) (1) \(\mathrm{abc}\) (2) acb (3) \(\mathrm{cab}\) (4) bca

Short answer

Question: Arrange the given steps in sequential order to find the LCM of 72 and 48 using the prime factorization method: (a) Find the prime factorization of 72 and 48. (b) Multiply the distinct prime factors with their highest exponents. (c) Identify all the distinct prime factors of both numbers. Answer: acb

Step-by-step solution

Prime Factorization of the Numbers

Find the prime factorization of 72 and 48. This is given as \(72=2^{3} \times 3^{2}\) and \(48=2^{4} \times 3^{1}\) in the exercise. Correct statement order: (a)

Identifying Distinct Prime Factors

Identify all the distinct prime factors of both numbers. In this case, the distinct factors are \(2^4\) and \(3^2\). Correct statement order: (c)

Finding the LCM using the Highest Powers of Prime Factors

Multiply the distinct prime factors with their highest exponents to find the LCM. The LCM in this case is \(2^{4} \times 3^{2}\). Correct statement order: (b) The correct sequential order of steps is: (1) (a) (2) (c) (3) (b) So the correct option among the given choices is \(\mathrm{acb}\).

Key concepts

Prime Factorization

Prime factorization is a technique used to express a whole number as a product of its prime factors. Prime numbers are the building blocks of all natural numbers and only have two divisors: one and themselves. To perform prime factorization, you begin by dividing the number by the smallest prime number possible (which is often 2), and continue the process with the quotient until you're left with a prime number.

For example, let's factorize the number 72. We start by dividing 72 by 2, the smallest prime number, getting 36. Again, 36 is divisible by 2, giving us 18. Continuing this way, we eventually break down 72 to \(2^3 \times 3^2\). Similarly, we can factorize 48 into \(2^4 \times 3\). Prime factorization lays the groundwork for finding the least common multiple (LCM) of numbers, as it clearly displays the prime numbers needed to construct each number.

Least Common Multiple

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of both or all of them. To put it simply, it's the smallest number into which all the given numbers can be divided without leaving a remainder.

Calculating the LCM is particularly useful when dealing with fractions, finding common denominators, and solving problems involving repeated events. The method of prime factorization greatly simplifies the calculation, as it allows us to easily compare the prime factors and their exponents to identify the required multiples.

Exponents in Prime Factorization

Exponents in prime factorization serve a significant role as they quantify the frequency a prime number is used as a factor in creating the original number. Instead of writing out a prime number multiple times, exponents provide a shorthand notation that is both efficient and clean.

For instance, rather than writing 2 x 2 x 2 x 3 x 3, we use exponents to condense this to \(2^3 \times 3^2\). Recognizing the highest exponents among the corresponding prime factors from numbers involved is critical for finding the LCM. If you compare the expressions for 72 (\(2^3 \times 3^2\)) and 48 (\(2^4 \times 3\)), the highest powers are 2 to the power of 4 from the second number and 3 to the power of 2 from the first. Thus, to find the LCM, we utilize these highest exponents, yielding \(2^4 \times 3^2\), which ensures that both original numbers can divide the LCM without leaving a remainder.