Problem 43 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. \(8,10,\) and 15

Step-by-Step Solution

Answer

The least common multiple is 120.

Step 1: Prime Factorization

Start by finding the prime factorization of each number. - **Prime factors of 8**: 8 = 2 × 2 × 2 = 2^3 - **Prime factors of 10**: 10 = 2 × 5 = 2^1 × 5^1 - **Prime factors of 15**: 15 = 3 × 5 = 3^1 × 5^1

Step 2: Determine the Highest Powers

Identify the highest power of each prime number that appears in the factorizations. - The highest power of **2** is 2^3 from the number 8. - The highest power of **3** is 3^1 from the number 15. - The highest power of **5** is 5^1 from both 10 and 15.

Step 3: Calculate the LCM

To find the least common multiple, multiply these highest powers together.\[LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5\]Calculate:- 8 × 3 = 24- 24 × 5 = 120

Step 4: Verify the Result

Ensure that 120 is divisible by each of the original numbers (8, 10, and 15) to verify correctness. - 120 ÷ 8 = 15 - 120 ÷ 10 = 12 - 120 ÷ 15 = 8 Since all quotients are integers, the LCM is verified.

Key Concepts

Prime Factorization

Prime factorization is a process of breaking down a number into a product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. By identifying the prime factors of a given set of numbers, we're setting the stage for finding their least common multiple (LCM).
For instance, in our original exercise:

• 8 can be expressed as a product of 2s: \(8 = 2 \times 2 \times 2 = 2^3\).
• 10 breaks down into two primes: \(10 = 2 \times 5 = 2^1 \times 5^1\).
• 15 is expressed using the primes 3 and 5: \(15 = 3 \times 5 = 3^1 \times 5^1\).

This step is crucial as it allows us to easily identify all the prime factors involved. Once you have the prime factorizations, you're halfway to understanding the mechanics of calculating the Least Common Multiple.

Highest Power

After determining the prime factors, the next step involves identifying the highest power of each prime that appears in any of the factorizations. This step ensures that the final LCM will be divisible by each of the original numbers, without unnecessary repetitions of lower power primes.

• The number 8 presents the highest power of 2, as \(2^3\).
• The number 15 introduces the prime number 3 in the highest power \(3^1\) since there are no other threes in our initial factorization.
• The highest power of 5, \(5^1\), appears twice (in 10 and 15), but since both are to the 1st power, it remains \(5^1\).

Collecting these highest powers means our LCM calculation has a solid base of all necessary prime factors to their fullest required extent.

Divisibility Verification

After calculating the Least Common Multiple, it is fundamental to verify the solution for accuracy. This step involves checking if the LCM is divisible by each of the original numbers.
This verification acts as a failsafe to ensure calculations were performed correctly. For our original problem, we determined the LCM to be 120. Here’s how you can confirm this:

• When divided by 8, \(120 \div 8 = 15\), which is an integer.
• Similarly, for 10, \(120 \div 10 = 12\), which is also an integer.
• And for 15, \(120 \div 15 = 8\), giving another integer.

Since all quotients are whole numbers, this confirms that 120 is indeed the smallest number divisible by 8, 10, and 15. Undertaking this verification step helps validate your calculations and gives peace of mind that the solution is accurate.