Problem 47 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. \(15,25,\) and 40

Step-by-Step Solution

Answer

The least common multiple of 15, 25, and 40 is 600.

Step 1: Prime Factorization of Each Number

First, prime factorize each number: - 15 can be factorized into primes as: \(15 = 3 imes 5\).- 25 can be factorized into primes as: \(25 = 5^2\).- 40 can be factorized into primes as: \(40 = 2^3 imes 5\).

Step 2: Identify All Unique Prime Factors

Identify all the distinct prime numbers present in the factorizations from Step 1: - The prime numbers are 2, 3, and 5.

Step 3: Determine the Highest Powers of Each Prime

For each prime number identified in Step 2, determine the highest power it is raised to in any of the numbers:- For 2, the highest power is \(2^3\) from 40.- For 3, the highest power is \(3^1\) from 15.- For 5, the highest power is \(5^2\) from 25.

Step 4: Calculate the Least Common Multiple (LCM)

Multiply the highest powers of each prime to find the LCM:\[LCM = 2^3 \times 3^1 \times 5^2 = 8 \times 3 \times 25 = 600\]

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as the product of its prime numbers. A prime number is one that has exactly two distinct positive divisors: 1 and itself. Prime factorization, therefore, involves finding which prime numbers multiply together to result in the original number. Let's see how this works with our given numbers. When you prime factorize 15, you find it is made up of two prime numbers: 3 and 5. So, we write this as:

• 15 = 3 × 5

For 25, the prime factorization is:

• 25 = 5 × 5 = 5²

Finally, for 40, the prime factorization results in:

• 40 = 2 × 2 × 2 × 5 = 2³ × 5

Prime factorization is the foundation step in finding the least common multiple (LCM), as it helps us understand the building blocks of each number.

Unique Prime Factors

In the context of prime factorization, unique prime factors refer to all the different prime numbers that appear in the factorization of any of the given numbers. This is important because they determine the factors we will use to compute the LCM. Consider the numbers we have:

• 15: prime factors are 3 and 5
• 25: prime factor is 5
• 40: prime factors are 2 and 5

After identifying the primes from each factorization, we collect all distinct ones. For 15, 25, and 40, the unique primes are:

• 2
• 3
• 5

Recognizing these unique prime factors helps us know which primes are essential in calculating the LCM, ensuring no important factor is missing.

Greatest Power of Primes

To find the least common multiple using prime factorization, an essential concept is determining the greatest power of each unique prime factor. This means identifying how many times each prime factor appears at its maximum in any of the numbers' prime factorizations. Let's consider our list of unique primes: 2, 3, and 5.

• For the prime number 2, the greatest number of times it appears is three times. So, we take 2³ from number 40.
• For the prime number 3, the greatest power is 3¹ from number 15.
• For the prime number 5, the greatest power is 5² from number 25.

We multiply these highest powers together to find the LCM, ensuring each prime factor is used in its most abundant form present. The LCM calculation is thus:\[LCM = 2^3 \times 3^1 \times 5^2 = 8 \times 3 \times 25 = 600\]Understanding the concept of greatest power of primes ensures that the LCM calculation takes into account the most sizeable collection of factors from each number, thus guaranteeing the smallest common multiple that all original numbers can equally divide.