Problem 52 - Prealgebra

Question

Find the least common multiple (LCM) of each set of numbers. $$45,30,35$$

Step-by-Step Solution

Answer

The LCM of 45, 30, and 35 is 630.

Step 1: Prime Factorization

Start by finding the prime factors of each number. For 45, the prime factorization is \( 3^2 \times 5 \). For 30, it is \( 2 \times 3 \times 5 \). For 35, it is \( 5 \times 7 \).

Step 2: Identify the Highest Power of Each Prime

Take each prime number that appears in the prime factorizations and find the highest power of each: \(2^1\), \(3^2\), \(5^1\) (because \(5\) appears once but \(3\) appears twice), and \(7^1\).

Step 3: Calculate the LCM

The LCM is obtained by multiplying the highest powers of all the primes together: \(LCM = 2^1 \times 3^2 \times 5^1 \times 7^1\).

Step 4: Perform the Multiplication

Calculate the multiplication: \(2 \times 9 \times 5 \times 7 = 630\). This value is the LCM.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a whole number into a set of prime numbers that, when multiplied together, give the original number. Prime numbers play a crucial role here. They are the building blocks of whole numbers, divisible only by 1 and themselves. This makes prime numbers essential in many areas of mathematics, including finding the least common multiple (LCM).
To prime factorize a number, start by dividing it by the smallest prime number (which is 2), and continue with larger primes as needed. For example:

• For the number 45, the prime factorization process involves dividing by 3 (a prime), yielding 15, then dividing 15 by 3 again, resulting in 5. The prime factors here are then represented as \(3^2 \times 5\).
• For 30, you would start dividing by 2, then follow with 3, and finally 5, rendering the factorization \(2 \times 3 \times 5\).
• For 35, since it is not divisible by the smaller prime numbers other than 5 and 7, you get \(5 \times 7\).

Prime factorization lays the groundwork for accurately determining the least common multiple by revealing all prime numbers involved.

Prime Numbers

Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. They are like the atoms of mathematics — small, indivisible numbers that compose larger numbers. Understanding prime numbers makes it easier to break down composite numbers through factorization.
When working with numbers to find their least common multiple, identifying the prime numbers involved is crucial. For instance:

• 2 is the smallest and the only even prime number.
• 3, 5, and 7 are the next few primes, sequentially.
• These numbers are repeatedly used in factorization as numbers are broken down into their prime components.

A clear grasp of prime numbers enables you to efficiently find which are new factors to consider when calculating the highest powers for the LCM.

Greatest Power Method

The Greatest Power Method is a strategic approach for finding the least common multiple (LCM) of a set of numbers. It revolves around using the prime factorization of each number to determine which powers of the primes should be included in the final LCM calculation. Here's how it works:
After you determine the prime factors of each number, the next step is selecting the highest power of each prime across all factorizations. This method ensures that the least common multiple is found by including each prime factor to the highest degree necessary to account for all numbers in the set.

• From the numbers 45, 30, and 35, the primes involved are \(2, 3, 5,\) and \(7\).
• Take the highest power of each: \(2^1\) from 30, \(3^2\) from 45, \(5^1\) since it appears across all but never squared for these, and \(7^1\) from 35.
• Multiplying these, the LCM is \(2^1 \times 3^2 \times 5^1 \times 7^1 = 630\).

This method ensures you cover all factors at their highest need, finding a product that represents the smallest shared multiple, helping you solve the LCM efficiently.