Problem 16 - Elementary Algebra

Question

For the following problems, find the least common multiple of given numbers. 7, 11, 33

Step-by-Step Solution

Answer

Answer: The least common multiple (LCM) of 7, 11, and 33 is 231.

Step 1: Find Prime Factors

First, we need to find the prime factors of each of the given numbers: 7, 11, and 33. The prime factorization of 7 is just 7 itself because 7 is prime: 7 → 7 The prime factorization of 11 is just 11 itself because 11 is prime: 11 → 11 The prime factorization of 33 can be found by dividing it by its smallest prime factor, 3, to get the quotient, 11. Since 11 is prime, we have found the prime factorization: 33 → 3 × 11

Step 2: Identify the Highest Powers of Each Prime Factor

Now that we have the prime factors of each number, determine the highest power of each prime factor that appears in any of the factorizations. This will help to ensure that our LCM is divisible by each of the given numbers. In our case, we have prime factors 7 and 11. We also have 3, which appears in the factorization of 33: 7 is present once. 11 is present once in the factor of 11 and also in the factor of 33. 3 is present once in the factor of 33. The highest powers are: 3^1, 7^1, and 11^1

Step 3: Calculate the Least Common Multiple

Finally, to find the LCM, we multiply the highest powers of each prime factor together: LCM = 3^1 × 7^1 × 11^1 Evaluating the expression, we get: LCM = 3 × 7 × 11 = 231 The least common multiple of the numbers 7, 11, and 33 is 231.

Key Concepts

Prime Factors

Prime factors are the basic building blocks of any number. They are primes, numbers greater than 1 only divisible by 1 and themselves. Identifying prime factors is essential to many mathematical procedures, including finding the Least Common Multiple (LCM).
In this exercise, the numbers given are 7, 11, and 33. Each of these can be broken down into their simplest prime factors:

• 7 is a prime number itself. No other numbers can divide it other than 1 and 7.
• 11 is also a prime number, sharing a similar property of only being divisible by 1 and 11.
• For 33, unlike 7 and 11, it is not a prime. Dividing 33 by the smallest prime number that fits, which is 3, results in 11. Both 3 and 11 are prime numbers, so 33 is neatly expressed as the product of these two: 3 × 11.

The identification of these prime factors, which are 7, 11, and 3, becomes crucial as we lay the groundwork to determine the LCM of the original numbers.

Prime Factorization

Prime factorization takes the task of identifying prime factors one step further by expressing a given number completely as a product of its prime factors.
This method helps provide a transparent view of what numbers multiply together to form the original number. For the current exercise with numbers 7, 11, and 33:

• 7: Since it's a prime number, its prime factorization is simply 7.
• 11: As with 7, the prime factorization is only 11 because it's singularly prime.
• 33: Here, the prime factorization unfolds as 3 × 11. Dividing by 3 offers 11, which is already prime, completing the factorization.

Prime factorization is immensely powerful because it breaks numbers down into fundamental components, facilitating easier calculations and further mathematical operations, leading gradually towards the resolution of complex problems like the LCM.

Highest Powers

Once prime factors are identified, the next step involves determining the highest powers of these factors present across all numbers. This ensures that the LCM includes a multiple of every individual number's factors.
For example, given the numbers in this exercise (7, 11, and 33), the prime factors 7, 11, and 3 appear as:

• 7: Appears in number 7 itself, to the power of 1 (7¹).
• 11: Is a factor of both 11 and 33. However, the highest power is still 1 (11¹).
• 3: Present only in 33 as a factor, so its highest power remains 1 (3¹).

Collectively, these highest powers, 3¹, 7¹, and 11¹, reflect the need to cover any occurrence of each prime factor within the given numbers, guaranteeing a common multiple.

Multiplication

The final step in determining the Least Common Multiple involves multiplying the highest powers of each prime factor together. This multiplication combines the strengths of each factor to form the smallest possible number that can evenly divide all given numbers.
In the problem at hand, after identifying the highest powers, we calculate:

• The expression is 3¹ × 7¹ × 11¹.
• This translates to the numerical multiplication: 3 × 7 × 11.
• Upon multiplication, the result is 231.

Thus, the Least Common Multiple of the numbers 7, 11, and 33 is conclusively 231. This value is vital because it represents the smallest shared product of these numbers' unique factors.