Problem 42 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. \(7,11,\) and 33

Step-by-Step Solution

Answer

The least common multiple is 231.

Step 1: List the Prime Factors

To find the least common multiple (LCM), we first need to list the prime factors of each number. - 7 is a prime number, so its only prime factor is 7. - 11 is a prime number, so its only prime factor is 11. - The number 33 can be factorized into 3 and 11 (since 3 x 11 = 33).

Step 2: Identify the Distinct Prime Factors

List all the distinct prime factors from the numbers 7, 11, and 33. They are 3, 7, and 11.

Step 3: Calculate LCM by Highest Powers

LCM is found by multiplying the highest powers of all distinct prime factors. - From 7, we take 7. - From 11, we take 11. - From 33, we take 3. So, LCM = 3 x 7 x 11 = 231.

Key Concepts

Understanding Prime Factors

When dealing with numbers, prime factors are the building blocks. A prime number is only divisible by 1 and itself, meaning it can't be broken down further.

• For example, the number 7 is prime because it has no divisors other than 1 and 7.
• Similarly, 11 is also prime with its only divisor being 1 and itself.
• On the other hand, 33 is not prime, but it can be expressed as a product of the prime numbers 3 and 11.

To find the prime factors, we keep dividing a number by its smallest prime until we can’t divide it any further. Establishing these factors is the first step toward calculating the least common multiple.

Steps in LCM Calculation

Calculating the least common multiple (LCM) of numbers involves combining their prime factors in the most effective way. This means using each distinct prime factor the greatest number of times it appears in any of the numbers. Here's how we calculate it:

• List all the prime factors of the numbers in question, acknowledging any repetition.
• For the numbers 7, 11, and 33, we've identified 7, 11, and 3 as necessary prime factors.
• Now, multiply these highest powers together to get the LCM. In this case, that's 3, 7, and 11.
• Multiplying these results in 231, which is the smallest number that all original numbers divide into without leaving a remainder.

The Role of Distinct Prime Factors

Distinct prime factors are unique building blocks derived from the numbers we are examining. By acknowledging only the essentials, you simplify LCM calculations considerably. Here's why they matter:

• Distinct prime factors include the unique primes from each number. Here, they are 3, 7, and 11.
• These distinct factors are important as they ensure we only multiply what's needed without redundancy.
• Including only unique primes once, we avoid unnecessary calculations and achieve the correct solution.
• This ensures the efficiency of finding the LCM. We only consider what's critical for all numbers to coexist under the same multiple.

This concept helps in maintaining focus on the truly essential numbers for our LCM calculation.