Problem 17 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers or monomials. $$16,3$$

Step-by-Step Solution

Answer

The least common multiple of 16 and 3 is 48.

Step 1: Understand the Definition of LCM

The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. In this case, we need to find the LCM of 16 and 3.

Step 2: Prime Factorization of Each Number

Factor both numbers into their prime components. For 16, the prime factorization is \(16 = 2^4\). For 3, the prime factorization is \(3 = 3^1\).

Step 3: Identify Maximum Powers for LCM

To find the LCM, take the highest power of each prime that appears in the factorizations. Here, we have primes 2 and 3. The maximum power of 2 is \(2^4\) from 16, and the maximum power of 3 is \(3^1\) from 3.

Step 4: Multiply the Highest Powers

The LCM is the product of these maximum prime powers. Thus, \(LCM = 2^4 \times 3^1\). Calculate this to get \(LCM = 16 \times 3 = 48\).

Key Concepts

Prime Factorization

Prime factorization is a helpful method to break down any integer into a set of prime numbers that multiply to form the original number. Prime numbers are numbers that are only divisible by 1 and themselves, like 2, 3, 5, 7, and so forth. Let's see how prime factorization is employed in our example of finding the least common multiple (LCM) of 16 and 3. For the number 16, prime factorization results in:

• 16 can be broken down into 2 x 8
• 8 can further be broken down into 2 x 4
• And 4 reduces to 2 x 2

Hence, when we multiply all the 2's together, we have:

• 2 x 2 x 2 x 2 = 2⁴

Similarly, for the number 3, since it is already a prime number, its prime factorization is simply:

• 3 = 3¹

Prime factorization not only simplifies the process of finding LCM but also makes it clearer which primes and their powers are necessary to achieve the LCM.

Divisible Numbers

Divisibility refers to one number being able to be divided by another without leaving a remainder. When finding the LCM, we aim to identify the smallest number that is divisible by all numbers in the set. For instance, if a number is said to be divisible by 3, this means when you divide that number by 3, you get a whole number. For example:

• 9 divided by 3 equals 3 (a whole number), so 9 is divisible by 3.

When we found the LCM of 16 and 3, the result was 48. This means 48 is the smallest number that both 16 and 3 can divide into without leaving any remainder. It serves as a universal number for these two specific numbers. To check:

• 48 divided by 16 equals 3
• 48 divided by 3 equals 16

Hence, the concept of divisibility helps confirm why the LCM is 48 by showing it can be exactly divided by both of the original numbers.

Multiples

Multiples are essentially the numbers you'll find when you multiply a number by any integer. They are infinitely possible, but some multiples are more prominent in mathematical solutions, like the least common multiple (LCM). Let's understand how multiples play out with our numbers 16 and 3. Multiples of 16 include:

• 16 (16 x 1)
• 32 (16 x 2)
• 48 (16 x 3)
• 64, and so on

Multiples of 3 are:

• 3 (3 x 1)
• 6 (3 x 2)
• 9 (3 x 3)
• 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, and so on

The least common multiple is the smallest number that appears in both lists. As you can see, the first number that appears in both lists of multiples is 48. This substantiates why the LCM is most often used for comparing multiple numbers to find a common denominator. With analytical approaches like identifying multiples, mathematical predictions and solutions become more tangible and understandable.