Problem 25 - Elementary Algebra

Question

For the following problems, find the least common multiple of given numbers. 8, 14, 28, 32

Step-by-Step Solution

Answer

Answer: The least common multiple of the numbers 8, 14, 28, and 32 is 224.

Step 1: Find the prime factors of each number

To find the prime factors of the numbers, we will divide the number by the smallest prime number that can evenly divide it. We will do this repeatedly until the quotient is 1. For 8: 8 = 2 × 2 × 2 For 14: 14 = 2 × 7 For 28: 28 = 2 × 2 × 7 For 32: 32 = 2 × 2 × 2 × 2 × 2

Step 2: Count the highest occurrences of each prime factor across all numbers

We will now count the highest occurrences of each prime factor in each of the given numbers: - For the prime factor 2: the highest count is 5 (in 32) - For the prime factor 7: the highest count is 1 (in 14 and 28)

Step 3: Calculate the LCM by multiplying the highest counts of each factor

The least common multiple (LCM) is calculated by multiplying the highest counts of each factor found in the previous step together. LCM = 2^5 × 7^1 = 32 × 7 = 224 The least common multiple of the given numbers 8, 14, 28, and 32 is 224.

Key Concepts

Prime Factorization

Understanding the prime factorization of a number is crucial when determining the least common multiple or greatest common divisor of a set of numbers. Prime factorization involves breaking down a composite number into the set of prime numbers that, when multiplied together, give back the original number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

Let's take the prime factorization of the number 8 from the exercise. To factorize 8, we start with the smallest prime number, which is 2, and continually divide until the quotient is 1:

• 8 ÷ 2 = 4
• 4 ÷ 2 = 2
• 2 ÷ 2 = 1

The prime factorization is therefore expressed as \( 8 = 2 \times 2 \times 2 \) or \( 2^3 \) to show that 2 is used three times as a factor. Following this straightforward method, we can find the prime factors for any composite number.

LCM Calculation

Once you have determined the prime factorization of all numbers involved, calculating the Least Common Multiple (LCM) is the next step. The LCM of a set of numbers is the smallest number that all of the numbers can divide into without a remainder. In other words, it is the smallest common multiple shared between the numbers.

The-step-by-step solution given for the numbers 8, 14, 28, and 32 demonstrates how to calculate the LCM algebraically after finding the prime factorization:

• Count the highest number of times each prime factor appears in any of the numbers’ prime factorizations.
• Multiply the prime factors together, each raised to the power of their highest count.

In our example, the prime factor 2's highest count is 5 and for prime factor 7 it's 1, so the LCM is \( 2^5 \times 7^1 \) which equals 224. This means 224 is the smallest number into which 8, 14, 28, and 32 can all evenly divide.

Divisibility Rules

Knowing divisibility rules helps to simplify the process of finding prime factors and subsequently the LCM. These rules provide quick ways to determine whether a number can be evenly divided by another without performing the actual division. For instance, a number is divisible by 2 if it is even, by 3 if the sum of its digits is divisible by 3, and by 5 if its last digit is 0 or 5.

These rules skip unnecessary calculations, allowing for a more efficient factoring process. For example, noticing that all the numbers in the exercise (8, 14, 28, and 32) end in an even digit, we can instantly conclude that they're all divisible by 2. If a number ends with a digit of 0, 4, or 8, it's divisible by 4, as are the given numbers 8, 28, and 32. Utilizing these rules effectively can significantly expedite the determination of prime factors and the calculation of the LCM.