Problem 82 - Fundamentals of Mathematics

Question

Find the least common multiple of each collection of numbers. \(40,50,\) and 180

Step-by-Step Solution

Answer

The least common multiple is 1800.

Step 1: Prime Factorization

Start by finding the prime factorization of each number: - 40 = 2^3 × 5 - 50 = 2 × 5^2 - 180 = 2^2 × 3^2 × 5

Step 2: Identify Maximum Powers

Identify the highest power of each prime that appears in the factorizations: - Prime 2: maximum power is 2^3 (from 40) - Prime 3: maximum power is 3^2 (from 180) - Prime 5: maximum power is 5^2 (from 50)

Step 3: Calculate LCM

Multiply the highest powers of all the prime factors together to find the LCM:\[\text{LCM} = 2^3 \times 3^2 \times 5^2\]

Step 4: Evaluate the Expression

Calculate the LCM using the expression from the previous step:- \(2^3 = 8\)- \(3^2 = 9\)- \(5^2 = 25\)- Then, \(8 \times 9 \times 25 = 1800\)

Step 5: Verify the Result

Verify the result by checking if 1800 is divisible by 40, 50, and 180: - 1800 ÷ 40 = 45 - 1800 ÷ 50 = 36 - 1800 ÷ 180 = 10 The remainders are zero, confirming that 1800 is the LCM.

Key Concepts

Prime Factorization

Prime factorization is a method where you express a number as a product of its prime numbers. A prime number is a number that is only divisible by 1 and itself, like 2, 3, 5, etc. Let's see how prime factorization works with 40, 50, and 180:

• For 40: Start dividing by the smallest prime number (2), and repeat the process until you are left with a prime number. - 40 ÷ 2 = 20 - 20 ÷ 2 = 10 - 10 ÷ 2 = 5 - As 5 is a prime number, the factorization is complete. So, 40 = 2^3 × 5.
• For 50: Apply the same method. - 50 ÷ 2 = 25 - 25 ÷ 5 = 5 - Thus, 50 = 2 × 5^2.
• For 180: Continue the process similarly. - 180 ÷ 2 = 90 - 90 ÷ 2 = 45 - 45 ÷ 3 = 15 - 15 ÷ 3 = 5 - Therefore, 180 = 2^2 × 3^2 × 5.

Prime factorization is a helpful technique as it breaks down the numbers into their basic building blocks. Understanding these factors aids in finding the least common multiple efficiently.

Highest Power of Primes

Finding the highest power of primes involves selecting the largest exponent for each prime number found in the factorization process. It assists in calculating the least common multiple (LCM) by ensuring all factors are included.
When finding the LCM, identify the highest power of each prime number from all factorizations. Here's how it works for our numbers:

• Prime 2: - Appears as 2^3 in the factorization of 40, - Therefore, the highest power is 2^3.
• Prime 3: - Only appears in 180 with 3^2, - Hence, 3^2 is selected as its maximum power.
• Prime 5: - The highest power comes from 50, which is 5^2.

Having the highest powers ensures the greatest inclusivity when multiplying the primes together to determine the LCM. This process guards against missing any necessary factors.

Divisibility Check

A divisibility check is crucial to confirm that the least common multiple is indeed the smallest number divisible by all given numbers. This step verifies the solution is correct.

• Why is the divisibility check important? It demonstrates the accuracy of your LCM calculation, ensuring the number holds true for all factors.
• How does it work? Simply divide the LCM by each of the original numbers. If there is zero remainder in each case, your calculation is verified correctly.

For the numbers 40, 50, and 180, the LCM is calculated as 1800:

• 1800 ÷ 40 = 45 with no remainder
• 1800 ÷ 50 = 36 with no remainder
• 1800 ÷ 180 = 10 with no remainder

This process ensures that each number divides perfectly into the LCM, confirming its correctness and reliability as the least common multiple. It's an essential step to validate your math work effectively.